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Following Barrow's idea of fractal black hole horizon, we re-derive black hole entropy of static spherically symmetric black holes. When a black hole absorbs matter its horizon area will increase. Given the spherically fractal structure, we conjecture that the minimal increase of the horizon area should be the area of the smallest bubble sphere. From this, we find the black hole entropy has a logarithmic form, which is similar to that of Boltzmann entropy if we consider $A/A_{pl}$ as the number of microscopic states. We further calculate temperatures and heat capacities of Schwarzschild, Reissner-Nordstr{\"o}m(RN), and RN-AdS black holes. It is found that their temperatures are all monotonically increasing and the heat capacities are all positive, which means these black holes are thermodynamically stable. Besides, for RN-AdS black hole we find its heat capacity has Schottky anomaly-like behavior, which may reflect the existence of the discrete energy level and restricted microscopical degree of freedom., Comment: 10 pages, 5 figures

We study the thermodynamic properties and critical behaviors of the topological charged black hole in AdS space under the consideration of the generalized uncertainty principle (GUP). It is found that only in the spherical horizon case there are Van der Waals-like first-order phase transitions and reentrant phase transitions. From the equation of state we find that the GUP-corrected black hole can have one, two and three apparent critical points under different conditions. However, it is verified by the Gibbs free energy that in either case there is at most one physical critical point., 13 pages, 9 figures, Advances in High Energy Physics, in press

On the basis of two requirements: the avoidance of the curvature singularity and the Maxwell theory as the weak field limit of the nonlinear electrodynamics, we find two restricted conditions on the metric function of $(2+1)$-dimensional regular black hole in general relativity coupled with nonlinear electrodynamics sources. By the use of the two conditions, we obtain a general approach to construct $(2+1)$-dimensional regular black holes. In this manner, we construct four $(2+1)$-dimensional regular black holes as examples. We also study the thermodynamic properties of the regular black holes and verify the first law of black hole thermodynamics., 13 pages, 4 figures. in press in PLB

In the framework of horizon thermodynamics, the field equations of Einstein gravity and some other second-order gravities can be rewritten as the thermodynamic identity: $dE=TdS-PdV$. However, in order to construct the horizon thermodynamics in higher-order gravity, we have to simplify the field equations firstly. In this paper, we study the fourth-order gravity and convert it to second-order gravity via a so-called " Legendre transformation " at the cost of introducing two other fields besides the metric field. With this simplified theory, we implement the conventional procedure in the construction of the horizon thermodynamics in 3 and 4 dimensional spacetime. We find that the field equations in the fourth-order gravity can also be written as the thermodynamic identity. Moreover, we can use this approach to derive the same black hole mass as that by other methods., Comment: 12 pages, no figure

The paper studies the development of the macroscopic methods of high-energy physics analysis. The authors consider the evolution of black holes within the phenomenological approach, analogous to classical thermodynamics, in which the black hole area determines its entropy, and the surface gravitation, correspondingly, - the temperature, in the framework of the relativist cosmological model (de Sitter universe). The research subject is the ways of calculation of effective thermodynamic properties of black holes. To calculate a black hole entropy, the authors apply the event horizon and cosmological horizon interdependence hypothesis. To accomplish the research task, the authors apply the system and structural-functional approaches, the methods of cosmology, relativistic mechanics and Einstein’s geometric theory of gravitation, in particular, the exact solutions of the Einstein field equations with the cosmological constant for the Reissner- Nordström metric for the space-time description. The authors find the analytical solution for the calculation of the total entropy of a spherically symmetric charged black hole in the Reissner- Nordström model for de Sitter universe. The paper shows that the expression for entropy includes not only the sum of entropies of the event horizon and cosmological horizon of the black hole, but also the additional term, taking into account their entanglement. The obtained results of black hole thermodynamics extend the analogy with the first law of thermodynamics, thus broadening the applicability of the approach to the cosmological studies.

Employing the fractional derivatives, we construct a more elaborate classification of phase transitions compared to the original Ehrenfest classification. In this way, a thermodynamic system can even undergo a fractional-order phase transition. We use this method to restudy the charged AdS black hole and Van der Waals fluids and find that at the critical point they both have a $4/3$-order phase transition, but not the previously recognized second-order one., Comment: 14 pages, 3 figures

Based on the consideration that the black hole horizon and the cosmological horizon of Reissner-Nordstr\"om black hole in de Sitter space are not independent each other, we conjecture the total entropy of the system should have an extra term contributed from the entanglement between the two horizons, except for the sum of the two horizon entropies. Making use of the globally effective first law and the effective thermodynamic quantities, we derive the total entropy and find that it will diverge as the two horizons tends to coincide., Comment: 3 pages, 2 figures. In the published version, there are some errors in the equations (13-15). We correct them in this version. The conclusion is not affected

We examine the entropy of black holes in de Sitter space and black holes surrounded by quintessence. These black holes have multiple horizons, including at least the black hole event horizon and a horizon outside it (cosmological horizon for de Sitter black holes and “quintessence horizon” for the black holes surrounded by quintessence). Based on the consideration that the two horizons are not independent each other, we conjecture that the total entropy of these black holes should not be simply the sum of entropies of the two horizons, but should have an extra term coming from the correlations between the two horizons. Different from our previous works, in this paper we consider the cosmological constant as the variable and employ an effective method to derive the explicit form of the entropy. We also try to discuss the thermodynamic stabilities of these black holes according to the entropy and the effective temperature.

In the quantized space-time of Snyder, generalized uncertainty relation and commutativity are both included. In this paper we analyze the possible form for the corrected Hawking temperature and derive it from the both effects. It is shown that the corrected Hawking temperature has a form similar to the one of noncommutative geometry inspired Schwarzschild black hole, however with an requirement $a/\sqrt {\theta }\leq 1/0.015$ for the noncommutative parameter 𝜃 and the minimal length a.

We demonstrate the existence of $P-V$ criticality of the topological Ho\v{r}ava-Lifshitz(HL) black holes with a spherical horizon $(k=1)$ in the extended phase space. With the electric charge, we find that the critical behaviors of the HL black hole are nearly the same as those of van der Waals(VdW) system. For the uncharged case, the HL black hole has a peculiar $P-V$ criticality. The critical behavior is completely controlled by a parameter $\epsilon$, but not the temperature $T$. When $\epsilon$ is larger than a critical value $\epsilon_c$, no matter what the temperature is, there will be the first-order phase transition. Moreover, we find that there is an infinite number of critical points which form a "critical curve". As far as we know, this is the first time to find this kind of peculiar $P-V$ criticality., Comment: 16 pages, 8 figures. some typos are corrected. a note is added at the last of the paper

Under two different metric ansatzes, the noncommutative geometry inspired black holes (NCBH) in the framework of Rastall gravity are derived and analyzed. We consider the fluid-type matter with the Gaussian-distribution smeared mass density. Taking a Schwarzschild-like metric ansatz, it is shown that the noncommutative geometry inspired Schwarzschild black hole (NCSBH) in Rastall gravity, unlike its counterpart in general relativity (GR), is not a regular black hole. It has at most one event horizon. After showing a finite maximal temperature, the black hole will leave behind a point-like massive remnant at zero temperature. Considering a more general metric ansatz and a special equation of state of the matter, we also find a regular NCBH in Rastall gravity, which has a similar geometric structure and temperature to that of NCSBH in GR., 12 pages, 5 figures. to match the published version

Based on the consideration that the black hole horizon and the cosmological horizon of Kerr-de Sitter black hole are not independent each other, we conjecture the total entropy of the system should have an extra term contributed from the correlations between the two horizons, except for the sum of the two horizon entropies. By employing globally effective first law and effective thermodynamic quantities, we obtain the corrected total entropy and find that the region of stable state for kerr-de Sitter is related to the angular velocity parameter $a$, i.e., the region of stable state becomes bigger as the rotating parameters $a$ is increases., Comment: 12 pages, 5 figures. arXiv admin note: text overlap with arXiv:1610.09886

Based on the work of Ghosh and Pereze, who view the black hole entropy as the logarithm of the number of quantum states on the Quantum Isolated Horizon (QIH)§ the entropy of Reissner—Nordstrom black hole is studied. According to the Unruh temperature, the statistical entropy of quantum fields under the background of Reissner—Nordstrom spacetime is calculated by means of quantum statistics. In the calculations we take the integral from the position of QIH to infinity, so the obtained entropy is the entanglement entropy outside the QIH. In Reissner—Nordstrom spacetime it is shown that if only the position of QIH is properly chosen the leading term of logarithm of the number of quantum states on the QIH is equal to the leading term of the entanglement entropy outside the black hole horizon, and both are the Bekenstein—Hawking entropy. The results reveal the relation between the entanglement entropy of black hole and the logarithm of the number of quantum states.

According to Maxwell’s equal area law we study the “real” phase diagram of the charged AdS black hole and the Kerr-AdS black hole. In the extended phase space constructed by treating the cosmological constant as pressure, the two kinds of AdS black hole display themselves like van der Waals system. Under the critical temperature Tc, there exists a regime where the condition of stable equilibrium will be violated. We employ the equal area law to find an isobar which is the real two phase coexistence line. As a byproduct we find a simple method to derive the critical value of the thermodynamic quantities.

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.

The weak field approximation of a model of de Sitter gauge theory of gravity is studied in two cases. Without torsion and spin current, the model cannot give the right non-relativistic approximation unless the density is a constant. With small torsion, a satisfactory Newtonian approximation can be obtained.

We study the thermodynamic stabilities of uncharged and charged black holes surrounded by quintessence (BHQ) by means of effective thermodynamic quantities. When the state parameter of quintessence $\omega_q$ is appropriately chosen, the structures of BHQ are something like that of black holes in de Sitter space. Constructing the effective first law of thermodynamics in two different ways, we can derive the effective thermodynamic quantities of BHQ. Especially, these effective thermodynamic quantities also satisfy Smarr-like formulae. It is found that the uncharged BHQ is always thermodynamically unstable due to negative heat capacity, while for the charged BHQ there are phase transitions of the second order. We also show that there is a great deal of difference on the thermodynamic properties and critical behaviors of BHQ between the two ways we employed., Comment: 18 pages, 13 figures. arXiv admin note: text overlap with arXiv:1605.08225

On the basis of horizon thermodynamics, we study the thermodynamic stability and $P-V$ criticality of topological black holes constructed in Ho\v{r}ava-Lifshitz (HL) gravity without the detailed-balance condition (with general $\epsilon$). In the framework of horizon thermodynamics, we do not need the concrete black hole solution (the metric function) and the concrete matter fields. It is shown that the HL black hole for $k=0$ is always thermodynamically stable. For $k=1$, the thermodynamic behaviors and $P-V$ criticality of the HL black hole are similar to those of RN-AdS black hole for some $\epsilon$. For $k=-1$, the temperature is classified into six types by their different features. Among them, we mainly focus on the type with triply degenerate thermodynamic state. It is also shown that there is a "thermodynamic singularity" for the $k=-1$ HL black hole, where the temperature and Gibbs free energy both diverge apart from a special pressure $P_s$., Comment: 10 pages, 11 figures

On the basis of horizon thermodynamics, we study the thermodynamic stability and $P-V$ criticality of topological black holes constructed in Ho��ava-Lifshitz (HL) gravity without the detailed-balance condition (with general $��$). In the framework of horizon thermodynamics, we do not need the concrete black hole solution (the metric function) and the concrete matter fields. It is shown that the HL black hole for $k=0$ is always thermodynamically stable. For $k=1$, the thermodynamic behaviors and $P-V$ criticality of the HL black hole are similar to those of RN-AdS black hole for some $��$. For $k=-1$, the temperature is classified into six types by their different features. Among them, we mainly focus on the type with triply degenerate thermodynamic state. It is also shown that there is a "thermodynamic singularity" for the $k=-1$ HL black hole, where the temperature and Gibbs free energy both diverge apart from a special pressure $P_s$., 10 pages, 11 figures

We obtain a magnetically charged regular black hole in general relativity. The source to the Einstein field equations is nonlinear electrodynamic field in a physically reasonable model of nonlinear electrodynamics (NED). "Physically" here means the NED model is constructed on the basis of three conditions: the Maxwell asymptotic in the weak electromagnetic field limit; the presence of vacuum birefringence phenomenon; and satisfying the weak energy condition (WEC). In addition, we analyze the thermodynamic properties of the regular black hole in two ways. According to the usual black hole thermodynamics, we calculate the heat capacity at constant charge, from which we know the smaller black hole is more stable. We also employ the horizon thermodynamics to discuss the thermodynamic quantities, especially the heat capacity at constant pressure., 12 pages, 5 figures

On the basis of horizon thermodynamics we study the thermodynamic stability of black holes constructed in general relativity and Gauss-Bonnet gravity. In the framework of horizon thermodynamics there are only five thermodynamic variables $E,P,V,T,S$. It is not necessary to consider concrete matter fields, which may contribute to the pressure of black hole thermodynamic system. In non-vacuum cases, we can derive the equation of state, $P=P(V,T)$. According to the requirements of stable equilibrium in conventional thermodynamics, we start from these thermodynamic variables to calculate the heat capacity at constant pressure and Gibbs free energy and analyze the local and global thermodynamic stability of black holes. It is shown that $P>0$ is the necessary condition for black holes in general relativity to be thermodynamically stable, however this condition cannot be satisfied by many black holes in general relativity. For black hole in Gauss-Bonnet gravity negative pressure can be feasible, but only local stable black hole exists in this case., 6 pages, 7 figures

By Maxwell equal area law we study the phase transition in higher dimensional Anti-de Sitter (AdS) Reissner-Nordstr\"{o}m (RN) black hole and Kerr black hole in this paper. The coexisting region of the two phases involved in the phase transition is found and the coexisting curves are shown in $P-T$ figures. We also analytically investigate the parameters which affect the coexisting curves. To better compare with ordinary thermodynamic systems, the Clapeyon equation is derived for higher dimensional AdS black holes. This paper can help to deepen the understanding of the phase transition and phase structure of AdS black holes., Comment: 21 pages, 16 figures

We study the phase transition of charged Gauss-Bonnet-de Sitter (GB-dS) black hole. For black holes in de Sitter spacetime, there is not only black hole horizon, but also the cosmological horizon. The thermodynamic quantities on the both horizons satisfy the first law of the black hole thermodynamics, respectively; moreover, there are additional connections between them. Using the effective temperature approach, we obtained the effective thermodynamic quantities of charged GB-dS black hole. According to Ehrenfest classification, we calculate some response functions and plot their figures, from which one can see that the spacetime undergoes a second-order phase transition at the critical point. It is shown that the critical values of effective temperature and pressure decrease with the increase of the value of GB parameter $\alpha$., Comment: 9 pages, 16 figures

In two frameworks, we discuss the thermodynamic stability of noncommutative geometry inspired Schwarzschild black hole (NCSBH). Under the horizon thermodynamics of black holes, we show that the NCSBH cannot be thermodynamically stable if requiring positive temperature. We note the inconsistency in the work of Larrañaga et al. and propose an effective first law of black hole thermodynamics for the NCSBH to eliminate the inconsistency. Based on the effective first law, we recalculate the heat capacity and the thermodynamic curvature by means of geometrothermodynamics (GTD) to revisit the thermodynamic stability.

There are both the black hole horizon and the cosmological horizon for the charged topological dilaton de Sitter spacetime. The thermodynamic quantities on both horizons satisfy the first law of black hole thermodynamics. Because all of these thermodynamic quantities depend on the mass $M$, the electric charge $Q$, and the cosmological constant $l$, the two horizons are not independent. Considering the connection between the black hole horizon and the cosmological horizon, we derive the effective thermodynamic quantities of the ($n+1$)-dimensional charged topological dilaton de Sitter spacetime. We find that the charged topological dilaton black hole in de Sitter spacetime has a similar phase transition and critical behavior to that in anti--de Sitter spacetime.

It is well known that there are black hole and the cosmological horizons for the Reissner-Nordstr\"{o}m-de Sitter spacetime. Although the thermodynamic quantities on the horizons are not irrelevant, they satisfy the laws of black hole thermodynamics respectively. In this paper by considering the relations between the two horizons we give the effective thermodynamic quantities in $(n+2)$-dimensional Reissner-Nordstr\"{o}m-de Sitter spacetime. The thermodynamic properties of these effective quantities are analyzed, moreover, the critical temperature, critical pressure and critical volume are obtained. We carry out an analytical check of Ehrenfest equations and prove that both Ehrenfest equations are satisfied. So the spacetime undergoes a second order phase transition at the critical point. This result is consistent with the nature of liquid--gas phase transition at the critical point, hence deepening the understanding of the analogy of charged dS spacetime and liquid--gas systems., Comment: 18 pages, 7 figures, 1 table

We study the $P-V/r_{+}$ criticality and phase transition of quantum-corrected black hole in asymptotic safety (AS) gravity in the extended phase space. For the black hole, the cosmological constant is dependent on the momentum cutoff or energy scale, therefore one can naturally treat it as a variable and connect it with the thermodynamic pressure. We find that for the quantum-corrected black hole there is a similar first-order phase transition to that of the van der Waals liquid/gas system. We also analyze the types of the phase transition at the critical points according to Ehrenfest's classification. It is shown that they are second-order phase transition., 10 pages,7 figures

It is wellknown that there are two horizons for the Reissner-Nordstrom-de Sitter spacetime, namely, the black hole horizon and the cosmological one. Both horizons can usually seem to be two independent thermodynamic systems; however, the thermodynamic quantities on both horizons satisfy the laws of black hole thermodynamics and are not independent. In this paper by considering the relations between the two horizons we give the effective thermodynamic quantities in Reissner-Nordstrom-de Sitter spacetime. The thermodynamic properties of these effective quantities are analyzed; moreover, the critical temperature, critical pressure, and critical volume are obtained. We also discussed the thermodynamic stability of Reissner-Nordstrom-de Sitter spacetime.

In this paper, we study the phase transition and the entropy spectrum of the Ba\~nados-Teitelboim-Zanelli (BTZ) black hole with torsion obtained in ($1+2$)-dimensional Poincar\'e gauge theory. By calculating the heat capacity, we find that the BTZ black hole with torsion we consider will experience phase transition at some critical point. This indicates that the critical behaviors of black holes do not only depend on the geometry of spacetime, but also have to do with the theory of gravity under consideration. In addition, we derive the entropy spectrum of the BTZ black hole according to the quasinormal modes and the adiabatic invariance. It shows that the area or entropy spectrum will also rely on the concrete gravitational action.

We show by explicit computations that there is a superficial inconsistency between the conventional first law of black hole thermodynamics and Bekenstein-Hawking area law for three types of regular black holes. The corrected form of the first law for these regular black holes is given. The derivation relies on the general structure of the energy-momentum tensor of the matter fields. When the black hole mass parameter $M$ is included in the energy-momentum tensor, the conventional form of the first law should be modified with an extra factor. In this case, the black hole mass $M$ can no longer be considered as the internal energy of the regular black holes., Comment: 11 pages, no figures, to be published in Classical and Quantum Gravity

We study the phase transition and the critical behavior of the BTZ black hole with torsion obtained in $(1+2)$-dimensional Poincar\'{e} gauge theory. According to Ehrenfest's classification, when the parameters in the theory are arranged properly the BTZ black hole with torsion may posses the second order phase transition which is also a smaller mass/larger mass black hole phase transition. Nevertheless, the critical behavior is different from the one in the van der Waals liquid/gas system. We also calculated the critical exponents of the relevant thermodynamic quantities, which are the same as the ones obtained in the Ho\v{r}ava-Lifshitz black hole and the Born-Infeld black hole., Comment: 15 pages, 5 figures. arXiv admin note: text overlap with arXiv:1310.1491

We study the equal area laws ofd-dimensional RN-AdS black hole. We choose two kinds of phase diagrams,P-VandT-S. We employ the equal area laws to find an isobar which is the real two-phase coexistence line. Our calculation is much simpler to derive the critical value of the thermodynamic quantities. According to the thermodynamic quantities, we also study the latent heat of the black hole.