Peculiar $P-V$ criticality of topological Ho\v{r}ava-Lifshitz black holes

Abstract

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.