Here we consider a flat FRW universe whose horizon entropy meets the Rényi entropy of nonextensive systems. In our model, the ordinary energy-momentum conservation law is not always valid. By applying the Clausius relation as well as the Cai-Kim temperature to the apparent horizon of a flat FRW universe, we obtain modified Friedmann equations. Fitting the model to the observational data on the current accelerated universe, some values for the model parameters are also addressed. Our study shows that the current accelerating phase of universe expansion may be described by a geometrical fluid, originated from the nonextensive aspects of geometry, which models a varying dark energy source interacting with the matter field in the Rastall way. Moreover, our results indicate that the probable nonextensive features of spacetime may also be used to model a varying dark energy source which does not interact with the matter field and is compatible with the current accelerated phase of the Universe.

Through the last years, it was demonstrated that quantum corrections of entropy, represented by logarithmic and power law corrections terms, constituted an association between semi-classical entropic areas and the curvature correction in Einstein–Hilbert’s Lagrangian and vice-versa. Loop quantum gravity approach provided the logarithmic corrections, which arises from quantum and thermal equilibrium fluctuations. On the other hand, Barrow’s entropy was introduced from the fact that the black hole surface can be modified due to quantum gravitational effects. The new exponent $$\Delta $$Δ that appears in Barrow’s entropy is a measure of this perturbation. In this letter we have analyzed the thermodynamical effects of the quantum fluctuations upon the geometry of a Barrow’s black hole. We demonstrated that new formulations of the equipartition law, which corresponds to the horizon energy, can be constructed from both entropic formalisms. Besides, we have calculated the heat capacity for both formulations and we discussed their thermal viability. We have also establish a condition on one of the constant pre-factors of the logarithmic correction.

In this paper we investigate the limits imposed by thermodynamics on a dark energy fluid. We obtain the heat capacities and the compressibilities for a dark energy fluid. The thermal and mechanical stabilities require these quantities to be positive. We show that dark energy fluids must satisfy the stability conditions and that such a requirement puts difficulties on the cosmic fluid models with negative constant equation-of-state (EoS) parameters. We also show that the observational constraints imposed by type Ia supernova, BAO and $H(z)$ data on a general dark energy fluid with a time-dependent EoS parameter are in conflict with the constraints imposed by thermodynamics. This result indicates that dark energy fluid models are unphysical.

Fluctuations can be incorporated into an equilibrium statistical mechanics setting through a superposition of different statistics, i.e., superstatistics. Herein, we have combined equipartition theorems arising out of superstatistics together with the holographic principle, to address the main consequences of fluctuations on gravitation and cosmology. We explored the effect of the three universality classes of superstatistics, namely, ${\chi }^2$, inverse ${\chi }^2$, and log-normal superstatistics, on the Jeans criterion and the Friedmann equations. We used some of the most recent cosmological data to constrain the universality parameter $q$ arising from these superstatistics. The dataset employed in our analysis provides $q>1$ at $1\sigma$ for the three superstatistics classes, indicating a deviation of the standard Maxwell-Boltzmann equilibrium statistics, $q=1$.