  Recently, Ho$\breve{r}$ava proposed a non-relativistic renormalizable theory
of gravity which is essentially a field theoretic model for a UV complete
theory of gravity and reduces to Einstein gravity with a non-vanishing
cosmological constant in IR. Also the theory admits a Lifshitz scale-invariance
in time and space with broken Lorentz symmetry at short scale. On the other
hand, at large distances higher derivative terms do not contribute and the
theory coincides with general relativity. Subsequently, Cai and his
collaborators and then Catiuo et al have obtained black hole solutions in this
gravity theory and studied the thermodynamic properties of the black hole
solution. In the present paper, we have investigated the black hole
thermodynamic for two choices of the entropy function - a classical and a
topological in nature. Finally, it is examined whether a phase transition is
possible or not.
