Correspondence of cosmology from non-extensive thermodynamics with fluids of generalized equation of state

We show that there is a correspondence between cosmology from non-extensive thermodynamics and cosmology with fluids of redefined and generalized equation of state. We first establish the correspondence in the case of basic non-extensive thermodynamics, and then we proceed by investigating the more consistent case, from the quantum field theoretical point of view, of varying exponent, namely depending on the scale. The obtained duality provides a way of explaining the complicated phenomenological forms of the effective fluid equation-of-state parameters that are being broadly used in the literature, since their microphysical origin may indeed lie in the non-extensive thermodynamics of spacetime. Finally, concerning the cosmological behavior, we show that at late times the effective fluid may drive the universe acceleration even in the absence of an explicit cosmological constant, and even if the initial fluid is the standard dust matter one. Similarly, at early times we obtain an effective cosmological constant which is enhanced through screening, and hence it can drive a successful inflation without spoiling the correct late-time acceleration.