Thermodynamics of $z=4$ Ho\v{r}ava-Lifshitz black holes

TL;DR

This paper investigates the thermodynamics of $z=4$ Hořava-Lifshitz black holes, revealing violations of the reverse isoperimetric inequality and identifying critical points with Van der Waals behaviors, challenging previous stability conjectures.

Contribution

It derives the first law and Smarr relation for $z=4$ Hořava-Lifshitz black holes, accounting for variable parameters, and demonstrates violations of the reverse isoperimetric inequality with thermodynamically stable cases.

Findings

Reverse isoperimetric inequality can be violated for various horizon geometries.

Black holes with hyperbolic horizons exhibit both Van der Waals and reverse Van der Waals behavior.

Positive specific heat at constant pressure and volume in cases violating the inequality.

Abstract

Thermodynamics of $z=4$ Ho\v{r}ava-Lifshitz black holes in 3+1 dimensions is studied in extended phase space. By using the scaling argument we find the Smarr relation and the first law for the black hole solutions of $z=4$ Ho\v{r}ava-Lifshitz gravity. We find that it is necessary to take into account the variation of dimensionful parameters of the theory in the first law. We find that the reverse isoperimetric inequality can be violated for spherical, flat, and hyperbolic horizons and in all such cases we have black holes for which the specific heat at constant pressure and volume are positive. This provides a counterexample to a recent conjecture stating that black holes violating the reverse isoperimetric inequality are thermodynamically unstable. We find for $z=4$ Ho\v{r}ava-Lifshitz black holes with hyperbolic horizons that there are two critical points: one showing Van der Waals behavior, the other reverse Van der Waals behavior.