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Aiming at a unified phase transition picture of the charged topological blackhole in Ho\v{r}ava-Lifshitz gravity, we investigate this issue not only incanonical ensemble with the fixed charge case but also in grand-canonicalensemble with the fixed potential case. We firstly perform the standardanalysis of the specific heat, the free energy and the Gibbs potential, andthen study its geometrothermodynamics. It is shown that the local phasetransition points not only witness the divergence of the specific heat, butalso witness the minimum temperature and the maximum free energy or Gibbspotential. They also witness the divergence of the corresponding thermodynamicscalar curvature. No matter which ensemble is chosen, the metric constructedcan successfully produce the behavior of the thermodynamic interaction andphase transition structure while other metrics failed to predict the phasetransition point of the charged topological black hole in former literature. Ingrand-canonical ensemble, we have discovered the phase transition which has notbeen reported before. It is similar to the canonical ensemble in which thephase transition only takes place when $k=-1$. But it also has its uniquecharacteristics that the location of the phase transition point depends on thevalue of potential, which is different from the canonical ensemble where thephase transition point is independent of the parameters. After an analyticalcheck of Ehrenfest scheme, we find that the new phase transition is a secondorder one. It is also found that the thermodynamics of the black hole inHorava-Lifshitz gravity is quite different from that in Einstein gravity.

A unified phase transition picture of the charged topological black hole in Horava-Lifshitz gravity