Black hole phase transitions in Ho\v{r}ava-Lifshitz gravity

We study black hole phase transitions in (deformed) Ho\v{r}ava-Lifshitz (H-L) gravity, including the charged black hole and KS black hole. Stability of black holes in H-L gravity and the phase transition signals are investigated. The Ricci scalar of the thermodynamical metrics of black holes in H-L gravity are calculated for the purpose of verifying or falsifying the correspondence between curvature singularities of thermodynamical metrics and phase transition signals proposed in [arXiv:1008.4482]. We find a probable counter-example to the correspondence that there is an infinite discontinuity at the specific heat curve for charged black hole with hyperbolic event horizon in H-L gravity but without a corresponding curvature singularity of thermodynamical metrics. The violation of the correspondence in H-L gravity may due to the non-spherical topology of its event horizon. We guess that the topology of the event horizon and the dimension of spacetime may influence the validity of the correspondence. Our results may provide new insights towards a better understanding of the H-L gravity, as well as black hole thermodynamics.