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

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.