Phase transition and thermodynamic stability of topological black holes in Ho\v{r}ava-Lifshitz gravity

On the basis of horizon thermodynamics, we study the thermodynamic stability and $P-V$ criticality of topological black holes constructed in Ho\v{r}ava-Lifshitz (HL) gravity without the detailed-balance condition (with general $ϵ$). In the framework of horizon thermodynamics, we do not need the concrete black hole solution (the metric function) and the concrete matter fields. It is shown that the HL black hole for $k=0$ is always thermodynamically stable. For $k=1$, the thermodynamic behaviors and $P-V$ criticality of the HL black hole are similar to those of RN-AdS black hole for some $ϵ$. For $k=-1$, the temperature is classified into six types by their different features. Among them, we mainly focus on the type with triply degenerate thermodynamic state. It is also shown that there is a “thermodynamic singularity” for the $k=-1$ HL black hole, where the temperature and Gibbs free energy both diverge apart from a special pressure $P_s$.