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The entanglement entropy of an acoustic black hole in a Bose—Einstein condensates (BEC) is derived, which is associated with the phonons generated via the Hawking mechanism in a sonic hole. Considering the dispersion relation of a BEC, we recalculate the entanglement entropy of the acoustic black hole by means of statistical method in two limits. We find that the entropy is still proportional to the area of event horizon, but with a coefficient dependent on the infrared cutoff.

Recently, Umetsu applied the tunneling method to the research on Hawking radiation of black holes using the dimensional reduction method near the horizon. In a simple way he obtained the radiation temperature of Kerr-Newman black hole. In this paper, using the dimensional reduction method near the horizon, we apply the tunneling method to the research on Hawking radiation of a general Kerr-de Sitter metrics in all dimensions. Our results support the views of Umetsu and extend the tunneling method proposed by Parikh and Wilczek.

We extend the method that Banerjee and Majhi have used to discuss Hawking radiation. Under the condition that the total energy and electrical charge of spacetime are conserved, we investigate Hawking radiation of the charged black hole by a new Tortoise coordinate transformation. Taking the reaction of the radiation of the particle to the spacetime into consideration, we not only derive the radiation spectrum that satisfies the unitary principle in quantum mechanics but also show that the contribution of ingoing particles is equal to the one of outgoing particles on the similar chemical potential term in radiation spectrum caused by charged particles.

After considering the generalized uncertainty principle, we discuss the quantum tunneling radiation of a five-dimensional Schwarzschild anti de Sitter black hole. The radiation spectrum and the correction value of the Bekenstein–Hawking entropy are derived. In a five-dimensional black hole the one order correction term in the Bekenstein–Hawking entropy correction term is proportional to the third power of the area, and the logarithmic correction term is a two-order small quantity. The correction term is related to the dimension constant introduced in the generalized uncertainty principle. Because the black hole entropy is not divergent, the lowest value of the five-dimensional Schwarzschild anti de Sitter black hole horizon radius is obtained. After considering the generalized uncertainty principle, the radiation spectrum is still consistent with normalization theory.

Utilizing the quantum statistical method and applying the new state density equation motivated by generalized uncertainty principle in quantum gravitaty, we avoid the difficulty in solving wave equation and directly calculate the partition function of bosonic and fermionic field on the background of rotating and charged black string. Then near the cosmological horizon, entropies of bosonic and fermionic field are calculated on the background of black string. When constant λ introduced in generalized uncertainty principle takes a proper value, we derive Bekenstein–Hawking entropy and the correction value corresponding cosmological horizon on the background of rotating and charged black string. Because we use the new state density equation, in our calculation there are not divergent term and small mass approximation in the original brick-wall method. From the view of quantum statistic mechanics, the correction value to Bekenstein–Hawking entropy of the black string is derived. It makes people deeply understand the correction value to the entropy of the black string cosmological horizon in non-spherical coordinate spacetime.

Considering corrections to all orders in Planck length on the quantum state density from a generalized uncertainty principle (GUP), we calculate the statistical entropy of the Bose field and Fermi field on the background of the four-dimensional spherically symmetric black holes without any cutoff. It is obtained that the statistical entropy is directly proportional to the area of horizon.

By using the entanglement entropy method, the statistical entropy of the Bose and Fermi fields in a thin film is calculated and the Bekenstein–Hawking entropy of Kerr–Newman black hole is obtained. Here, the Bose and Fermi fields are entangled with the quantum states in Kerr–Newman black hole and are outside of the horizon. The divergence of brick-wall model is avoided without any cutoff by the new equation of state density obtained with the generalized uncertainty principle. The calculation implies that the high density quantum states near the event horizon are strongly correlated with the quantum states in black hole. The black hole entropy is a quantum effect. It is an intrinsic characteristic of space–time. The ultraviolet cutoff in the brick-wall model is unreasonable. The generalized uncertainty principle should be considered in the high energy quantum field near the event horizon. From the calculation, the constant λ introduced in the generalized uncertainty principle is related to polar angle θ in an axisymmetric space–time.

Recently, the Hawking radiation of a black hole has been studied using the tunnel effect method. The radiation spectrum of a black hole is derived. By discussing the correction to spectrum of the rotating black hole, we obtain the canonical entropy. The derived canonical entropy is equal to the sum of Bekenstein–Hawking entropy and correction term. The correction term near the critical point is different from the one near others. This difference plays an important role in studying the phase transition of the black hole. The black hole thermal capacity diverges at the critical point. However, the canonical entropy is not a complex number at this point. Thus we think that the phase transition created by this critical point is the second order phase transition. The discussed black hole is a five-dimensional Kerr-AdS black hole. We provide a basis for discussing thermodynamic properties of a higher-dimensional rotating black hole.

Recently, there has been much attention devoted to resolving the quantum corrections to the Bekenstein–Hawking black hole entropy. The different correction leading terms are obtained by the different methods. In this paper, we calculate the correction to SAdS5 black hole thermodynamic quantity due to the generalized uncertainty principle. Furthermore we derive that the black hole entropy obeys Bekenstein–Hawking area theorem. The entropy has infinite correction terms. And every term is finite and calculable. The corrected Cardy–Verlinde formula is derived. In our calculation, Bekenstein–Hawking area theorem still holds after considering the generalized uncertainty principle. We have not introduced any hypothesis. The calculation is simple. Physics meaning is clear. We note that our results are quite general. It is not only valid for four-dimensional spacetime but also for higher-dimensional SAdS spacetime.

By using the entanglement entropy method, the statistical entropy of the Bose and Fermi fields in a thin film is calculated and the Bekenstein–Hawking entropy of six-dimensional Horowitz–Strominger black hole is obtained. Here, the Bose and Fermi fields are entangled with the quantum states in six-dimensional Horowitz–Strominger black hole and the fields are outside of the horizon. The divergence of brick-wall model is avoided without any cutoff by the new equation of state density obtained with the generalized uncertainty principle. The calculation implies that the high density quantum states near the event horizon are strongly correlated with the quantum states in black hole. The black hole entropy is a quantum effect. It is an intrinsic characteristic of space–time. The ultraviolet cutoff in the brick-wall model is unreasonable. The generalized uncertainty principle should be considered in the high energy quantum field near the event horizon. Using the quantum statistical method, we directly calculate the partition function of the Bose and Fermi fields under the background of the six-dimensional black hole. The difficulty in solving the wave equations of various particles is overcome.

Recently, there has been much attention devoted to resolving the quantum corrections to the Bekenstein-Hawking black hole entropy. In particular, many researchers have expressed a vested interest in the coefficient of the logarithmic term of the black hole entropy correction term. In this paper, we calculate the correction value of the black hole entropy by utilizing the generalized uncertainty principle and obtain the correction term caused by the generalized uncertainty principle. Because in our calculation we think that the Bekenstein-Hawking area theorem is still valid after considering the generalized uncertainty principle, we derive that the coefficient of the logarithmic term of the black hole entropy correction term is positive. This result is different from the known result at present. Our method is valid not only for single horizon spacetime but also for spin axial symmetric spacetimes with double horizons. In the whole process, the physics idea is clear and calculation is simple. It offers a new way for studying the entropy correction of the complicated spacetime.

The generalized uncertainty relation is introduced to calculate entropy of the black hole. By using quantum statistical method, we directly obtain the partition function of Bose and Fermi field on the background of the plane symmetry black hole. Then we calculate the entropy of Bose and Fermi field on the background of black hole near the horizon of the black hole. In our calculation, we need not introduce cutoff. There are not the left out term and the divergent logarithmic term in the original brick-wall method. And it is obtained that the entropy of the black hole is proportional to the area of the horizon. The inherent contact between the entropy of black hole and the area of horizon is opened out. Further it is shown the entropy of black hole is entropy of quantum state on the surface of horizon. The black hole’s entropy is the intrinsic property of the black hole. The entropy is a quantum effect.

Recently, Hawking radiation of the black hole has been studied using the tunnel effect method. It is found the radiation spectrum of the black hole is not a strictly pure thermal spectrum. How the departure from pure thermal spectrum affects the entropy? This is a very interesting problem. In this paper, we calculate the partition function by energy spectrum obtained by tunnel effect. Using the relation between the partition function and entropy, we derive the expression of entropy the general charged black hole. In our calculation, we not only consider the correction to the black hole entropy due to fluctuation of energy but also consider the effect of the change of the black hole charges on entropy. We discuss Reissner-Nordstrom black hole and obtain that Reissner-Nordstrom black hole cannot approach the extreme black hole by changing its charges.

Recently, there has been much attention devoted to resolving the quantum corrections to the Bekenstein–Hawking black hole entropy. In particular, many researchers have expressed a vested interest in the coefficient of the logarithmic term of the black hole entropy correction term. In this Letter, we calculate the correction value of the black hole entropy by utilizing the generalized uncertainty principle and obtain the correction terms of entropy, temperature and energy caused by the generalized uncertainty principle. We calculate Cardy–Verlinde formula after considering the correction. In our calculation, we only think that the Bekenstein–Hawking area theorem is still valid after considering the generalized uncertainty principle and do not introduce any assumption. In the whole process, the physics idea is clear and calculation is simple. It offers a new way for studying the corrections caused by the generalized uncertainty principle to the black hole thermodynamic quantity of the complicated spacetime.

In this paper we discuss thermodynamics parameters of black hole horizon and cosmological horizon in general high-dimensional space-time. We obtain that the entropy of a cosmological horizon can be described by the Cardy-Verlinde formula. However, the entropy of black hole horizon will be expressed in a form of the Cardy-Verlinde formula, if one adopts the methods given by Abbott and Deser to compute the mass of a black hole in general high-dimensional space-time. Through discussion, relation among various thermodynamics parameters of the black hole in general high-dimensional space-time is given. That is, differential formula of the first law of thermodynamics is obtained. Because we discuss the general high-dimensional space-time, our result has universality.

The generalized uncertainty relation is introduced to calculate quantum statistic entropy of a black hole. By using the new equation of state density motivated by the generalized uncertainty relation, we discuss entropies of Bose field and Fermi field on the background of the five-dimensional spacetime. In our calculation, we need not introduce cutoff. There is not the divergent logarithmic term as in the original brick-wall method. And it is obtained that the quantum statistic entropy corresponding to black hole horizon is proportional to the area of the horizon. Further it is shown that the entropy of black hole is the entropy of quantum state on the surface of horizon. The black hole's entropy is the intrinsic property of the black hole. The entropy is a quantum effect. It makes people further understand the quantum statistic entropy.

By using the method of quantum statistics and via the new equation of state density motivated by the generalized uncertainty relation in the quantum gravity, we avoid the difficulty of solving wave equation, and directly derive the partition functions of bosonic and fermionic field in Toroidal black hole. Then we calculate the entropies of bosonic and fermionic field near the black hole horizon on the background of a black hole. In our result, the divergent logarithmic term and ultraviolet cutoff in the original brick-wall method by which the black hole entropy was derived no longer exist. It is shown that the entropy of the black hole is proportional to the area of the horizon. These results are similar to that in the black hole with horizon topology S2.

By using the method of quantum statistics, we derive directly the partition functions of bosonic and fermionic field in the d-dimensional Horowitz-Strominger black hole. The statistical entropy of black hole is obtained by an improved brick—wall method. When we choose proper parameter in our results, we can obtain that the entropy of the black hole is proportional to the area of the horizon. In our result, there don't exist the left out term and divergent logarithmic term given in the original brick—wall method. We avoid the difficulty in solving the wave equation of scalar and Dirac field. And we offer a simple and direct way of studying entropy of the higher-dimensional complicated black hole.

The properties of the thermal radiation are discussed by using the new equation of state density motivated by the generalized uncertainty relation in the quantum gravity. There is no burst at the last stage of the emission of dilatonic black hole. When the new equation of state density is utilized to investigate the entropy of a bosonic field and fermionic field outside the horizon of a static dilatonic black hole, the divergence appearing in the brick wall model is removed, without any cutoff. It is derived from the contribution of the vicinity of the horizon that the entropy is proportional to the horizon area.

A Horowitz–Strominger black hole is discussed through a new equation of state density motivated by the generalized uncertainty relation in quantum gravity. There is no burst in the last stage of emission from a Horowitz–Strominger black hole. When the new equation of state density is used to investigate the entropy of bosonic field and fermionic field outside the horizon of a static Horowitz–Strominger black hole, the divergence that appears in the brick-wall model is removed without any cutoff. The entropy proportional to the horizon area is derived from the contribution in the vicinity of the horizon.

Improving the membrane model by which the entropy of the black hole is studied, we study the entropy of the black hole in the non-thermal equilibrium state. To give the problem stated here widespread meaning, we discuss the (n+2)-dimensional de Sitter spacetime. Through discussion, we obtain that the black hole's entropy which contains two horizons (a black hole's horizon and a cosmological horizon) in the non-thermal equilibrium state comprises the entropy corresponding to the black hole's horizon and the entropy corresponding to the cosmological horizon. Furthermore, the entropy of the black hole is a natural property of the black hole. The entropy is irrelevant to the radiation field out of the horizon. This deepens the understanding of the relationship between black hole's entropy and horizon's area. A way to study the bosonic and fermionic entropy of the black hole in high non-thermal equilibrium spacetime is given.

In this paper, by using quantum statistical method, we obtain the partition function of Bose field and Fermi field on the background of the 5-dimensional rotating black hole. Then via the improved brick-wall method and membrane model, we calculate the entropy of Bose field and Fermi field of the black hole. And it is obtained that the entropy of the black hole is not only related to the area of the outer horizon but also is the function of inner horizon's area. In our results, there are not the left out term and the divergent logarithmic term in the original brick-wall method. The doubt that why the entropy of the scalar or Dirac field outside the event horizon is the entropy of the black hole in the original brick-wall method does not exist. The influence of spinning degeneracy of particles on entropy of the black hole is also given. It is shown that the entropy determined by the areas of the inner and outer horizons will approach zero, when the radiation temperature of the black hole approaches absolute zero. It satisfies Nernst theorem. The entropy can be taken as the Planck absolute entropy. We provide a way to study higher dimensional black hole.

Properties of the thermal radiation of black holes are discussed using a new equation of state density motivated by the generalized uncertainty relation in quantum gravity. There is no burst at the last stage of emission from a spherically symmetric black hole. When the new equation of state density is used to investigate the entropy of a bosonic field and fermionic field outside the horizon of a static spherically symmetric black hole, the divergence that appears in the brick-wall model is removed without any cutoff. The entropy proportional to the horizon area is derived from the contribution from the vicinity of the horizon.

By using the method of quantum statistics, we derive directly the partition functions of bosonic and fermionic fields in the N-dimensional spherically symmetric charged black hole space-time. The statistical entropy of black hole is obtained by an improved brick-wall method. When we choose proper parameters in our results, we can obtain that the entropy of black hole is proportional to the area of horizon. In our result, there do not exist neglected term and divergent logarithmic term given in the original brick-wall method. We avoid the difficulty in solving the wave equation of scalar and Dirac fields. We offer a simple and direct way of studying entropy of the higher-dimensional black hole.

By using the method of quantum statistics, we directly derive the partition functions of bosonic and fermionic field in Kaluza—Klein black hole with axial symmetry. Then via the improved brick-wall method, membrane model, we obtain that the entropy of bosonic and fermionic field in black hole is proportional to the area of horizon. In our result, the stripped term and the divergent logarithmic term no longer exist. The problem that the state density is divergent around the horizon doesn't exist either. We also give the influence of the spining degeneracy of particles on the entropy of black hole. We offer a new, simple, and direct way of calculating the entropy of different complicated black holes.

By using the method of quantum statistics, we derive the partition function of bosonic and fermionic field in various coordinates and obtain the integral expression of the entropy of a black hole. Then via the improved brick-wall method, membrane model, we obtain that if we choose proper parameter, the entropy of black hole is proportional to the area of horizon. In our result, the stripped term and the divergent logarithmic term in the original brick-wall method no longer exist. We offer a new simple and direct way of calculating the entropy of black holes in various coordinates.

By making use of the method of quantum statistics, we directly derive the partition function of bosonic and fermionic fields in Reissner–Nordstrom–De Sitter black hole and obtain the integral expression of black hole's entropy and the entropy to which the cosmic horizon surface corresponds. It avoids the difficulty in solving the wave equation of various particles. Then via the improved brick-wall method, i.e. the membrane model, we calculate black hole's entropy and cosmic entropy and find out that if we let the integral upper limit and lower limit both tend to the horizon, the entropy of black hole is proportional to the area of horizon and the entropy to which cosmic horizon surface corresponds is proportional to the area of cosmic horizon. In our result, the stripped term and the divergent logarithmic term in the original brick-wall method no longer exist. In the whole process, the physical idea is clear and the calculation is simple. We offer a new simple and direct way for calculating the entropy of different complicated black holes.

By using the method of quantum statistics, we directly derive the partition function of bosonic and fermionic field in dilatonic black hole and obtain the integral expression of the black hole's entropy, which avoids the difficulty in solving the wave equationof various particles. Then via the improved brick-wall method, membrane model, we obtain that we can choose proper parameter in order to let the thickness of film tend to zero and have it approach the surface of its horizon. Consequently the entropy of the black hole is proportional to the area of its horizon. In our result, the stripped term and the divergent logarithmic term in the original brick-wall method no longer exist. In the whole process, physics idea is clear; calculation is simple. We offer a new simple and direct way of calculating the entropy of different complicated black holes.

By using the method of quantum statistics, we directly derive the partition function of bosonic and fermionic field in Reissner-Nordstrom-anti-de Sitter black hole and obtain the integral expression of black hole's entropy. It avoids the difficulty in solving the wave equation of various particles. Then via the improved brick-wall method, membrane model, we calculate the statistical entropy of a film with the thickness of (N - 1)ξ around the outside of horizon. In our result we can choose proper parameter in order to let the thickness of film tend to zero and have it approach the surface of horizon. Consequently, the entropy of black hole is proportional to the area of horizon. The stripped term and the divergent logarithmic term in the original brick-wall method no longer exist. In the whole process, physics idea is clear; calculation is simple. We offer a new simple and direct way of calculating the entropy of different complicated black holes.

By using the method of quantum statistics, we directly derive the partition function of bosonic and fermionic field in Reissner–Nordstrom black hole. It is the same as the one obtained by solving wave equation approximately. Then via the improved brick-wall method, membrane model, obtain that the entropy of bosonic and fermionic field in black hole is proportional to the area of horizon. In our result, the stripped term and the divergent logarithmic term don't exist any more. The problem that the state density is divergent around the horizon doesn't exist either. We also give the influence factor of the spining degeneracy of various particles on the entropy of black hole. In the whole process, we do not take any approximation. We offer a new simple and direct way of calculating the entropy of different complicated black holes.

The free energy and the entropy of scalar field are calculated by brick-wall in the axisymmetric Einstein–Maxwell–Dilaton-axion black hole. It is shown that when the black hole has inner and outer horizons, the entropy is not only related to the area of an outer horizon but also is the function of the area of an inner horizon. When the area of an inner horizon approaches zero, we can obtain the known conclusion. The entropy expressed by location parameter of outer horizon and inner horizons approaches absolute zero. It obeys Nernst theorem. It can be taken as Planck absolute entropy of a black hole.

Under nonspherical coordinates, the free energy and the entropy of a scalar field are calculated in terms of the brick-wall method on the background of a nonasymptotically flat cylindrical black hole. It is shown that the entropy is not only related to the area of an outer horizon but also function of inner horizon at nonasymptotically flat space–time, when a black hole has both inner and outer horizons. Further, the entropy expressed by location parameter of outer and inner horizons approaches zero, when the radiation temperature of a black hole approaches zero. It satisfies the Nernst theorem and can be taken as Planck absolute entropy of a black hole.

The difficulty of calculating energy-momentumtensors is avoided by finding directly the solution ofKlein–Gordon and Dirac equations near the horizon.Both the location of the event horizon and the Hawking radiation temperature of a nonstatic chargedblack hole are shown. The results indicate that theHawking radiation temperature can be regarded as acompensating effect under the timescaletransformation.

The existence of the Hawking radiation of theblack hole surely affects spacetime. Here, using athermodynamic approach, to avoid the difficulty offinding the energy-momentum tensor, we obtainexpressions for the energy and entropy of theReissner–Nordstrom black hole (RNBH), and providea better way of solving the backreaction of more complexblack holes.

We enlarge the range of usage of the research of the non-static black-hole quantum thermal effect, which differs from both steady and advanced Eddington coordinate V state, get the solution of the Dirac equation near an evaporating charged black hole, obtain event horizon location and radiation temperature and spectrum of the particle, find a property attributed to this kind of black hole. Its local event horizon consists of two close event horizon surfaces. Its temperature is determined by both event horizons.

In this paper, with the thermodynamical method first invented by a Chinese scholar, we research the Kerr-Newmann space-time. Having considered the back-reaction, we disuss the energy, the angular momentum and the revised expression of entropy of the Kerr-Newmann black hole (KNBH). Up to now, no one has obtained this result through other methods.

Both the event horizon equation and the radiation temperature of a rotating and arbitrarily accelerating non-stationary black hole are shown. The event horizon surface is no longer a sphere. The temperature and the spectral distribution of radiation depend on not only the time, but also the direction angles.

The partition functions of bosonic and fermionic fields in Horowitz–Strominger black hole are derived directly by quantum statistical method. Then via the improved brick-wall method (membrane model), the statistical entropy of black hole is obtained. If a proper parameter is chosen in our result, it is found out that the entropy is proportional to the area of horizon. The stripped term and the divergent logarithmic term in the original brick-wall method no longer exist. The difficulty in solving the wave equations of scalar and Dirac fields is avoided. A new neat way of calculating the entropy of various complicated black holes is offered.

We extend the classical Damour–Ruffini method and discuss Hawking radiation spectrum of high-dimensional rotating black hole using Tortoise coordinate transformation defined by taking the reaction of the radiation to the spacetime into consideration. Under the condition that the energy and angular momentum are conservative, taking self-gravitation action into account, we derive Hawking radiation spectrums which satisfy unitary principle in quantum mechanics. It is shown that the process that the black hole radiates particles with energy ω is a continuous tunneling process. We provide a theoretical basis for further studying the physical mechanism of black-hole radiation.

We extend the classical Damour-Ruffini method and discuss Hawking radiation in Schwarzschild -de Sitter black hole. Under the condition that the total energy of spacetime is conservative, considering the effect of radiation particle on the spacetime and the relevance between the black hole event horizon and cosmological horizon, we obtain the black hole radiation spectrum. The radiation spectrum is no longer a pure thermal spectrum. It is related to Bekenstein-Hawking entropy change corresponding to the black hole event horizon and cosmological horizon. The result is consistent with an underlying unitary theory.

Applying the new equation of state density motivated by the generalized uncertainty relation in the quantum gravity field，we investigate the black hole entropy on the background of three-dimensionnal BTZ. When the λ with the Planck scale，and relative to the dimensions of spacetime introduced in generalized uncertainty relation has a fixed value，we obtain the Bekenstein-Hawking entropy of BTZ black hole and the correction term. Because we have used the new equation of state density，in our results the divergence term appearing in the brick-wall model is removed，at the same time with holding the small mass approximation. Thus the correction value of the Bekenstein-Hawking entropy of the black hole is derived from the quantum statistical view. It depends the understanding of the correction value of the black hole entropy.

Using the quantum statistical method, the difficulty of solving wave equation on the background of the black hole is avoided. We directly solve the partition functions of bosonic field and fermionic field on the background of the axisymmetric Kerr black hole through using the new equation of state density motivated by the generalized uncertainty relation in the quantum gravity theory. Then the entropy of the bosonic field and fermionic field near the horizon of the black hole are calculated. In our results the divergence appearing in the brick wall model is removed, as well as without using the small mass approximation. The series expression of the statistical entropy of the black hole is convergent. Therefore, it gives a better understanding of the black hole statistical entropy in non-spherical symmetry spacetimes.

Taking into consideration the radiation particle reaction with the total energy of spacetime conservation, Hawking radiation from black hole is re-investigated using the Damour-Ruffini method. It is found that the tunneling probability relates to Bekenstein-Hawking entropy, and the radiation spectrum is not exactly thermal. The result is consistent with the works of other researchers， and satisfies the unitary theory of quantum mechanics.

Recently, Hawking radiation of the black hole was studied by tunnel effect method. It was found that the radiation spectrum of the black hole is not a strictly pure thermal spectrum. How the deviation between the radiation spectrum from pure thermal spectrum affect the entropy is a very interesting problem. In this paper, we calculate the partition function using the energy spectrum derived by tunnel effect, then obtain the entropy of the black hole. When we take first order approximation, the logarithmic term of entropy correction is consistent with the one considering the generalized uncertainty relation. However, in our calculation, there is no uncertain factor. And the coefficient of logarithmic correction term is the same one after considering the correction to the black hole entropy due to the thermal fluctuation. There are no divergent terms in our result when the thermal capacity of the black hole is negative. So our discussion has universality.

When we take the black hole as a canonical ensemble composed of a naked black hole and the two-dimensional thermodynamic surface (horizon of the black hole), using the quantum statistical method, we derive the energy spectrum of the black hole Hawking radiation. The relation between the radiation temperature of the black hole and the entropy is obtained. This gives an insight into the process of the black hole Hawking radiation.

Using the quantum statistical method, we directly obtain the partition function of Bose field and Fermi field on the background of Schwarzschild spacetime. The integral expression is obtained. According to the latest research results, Hawking radiation process of the black hole is a tunnel effect. During the creation of tunnel effect, the energy of the black hole changes. The lower limit of integral is the location of the black hole horizon. Therefore, the leading term of the black hole entropy is a quarter of the horizon area. In our calculation, there are no cutoffs. We obtain that the energy of the black hole radiation particle is proportional to the radiation temperature. This provides an insight into the thermodynamics and quantum property of the black hole.

In this paper, in the two-parameter five-dimensional(5D) rotating black hole, we discuss the thermodynamic parameters corresponding to the black hole's horizon and the thermodynamic parameters corresponding to the cosmological horizon. It is obtained that the entropy of cosmological horizon can be written in the form of Cardy-Verlinde formula. However, if we want to write the entropy of the bla ck hole's horizon in the form of Cardy-Verlinde formula, we have to use the met hods offered by Abbott and Deser to calculate the mass of the 5D black hole with double- spinning parameter. Through the research, we can give the relation among the thermodynamic parameters of the 5D black hole, that is, the differential ex pression of the first law of thermodynamics.