Thermodynamical phase transitions of AdS black hole in Horava-Lifshitz gravity

In this paper, we investigate the thermodynamical phase transitions of the spherically symmetric AdS black hole in Horava-Lifshitz (HL) gravity for the detailed balance violation parameter 0 <=epsilon 2 <= 1. First, by calculating the HP temperature THP, the minimum temperature Tmin and the Gibbs free energy G of the AdS black hole in different values of parameter epsilon, we find that the HP phase transition can occur at all pressures. Comparing the HP temperature THP with the minimum temperature Tmin, we notice that the ratio of THP/Tmin tends to 1 when d ->infinity. Second, for the small-large black hole phase transition of the AdS black hole, we discover that the black hole heat capacity is divergent at the minimum temperature Tmin, the small-large black hole phase transition occurs at the same time. Furthermore, the black hole entropy also exists SHP when T=THP, and Smin when the black hole temperature has its minimum Tmin, we surprisingly find that the ratio of SHP/Smin tends to e when the dimension d ->infinity, which provides a general relation between the Hawking-Page phase transition and the small-large black hole phase transition of the AdS black hole in the large dimension. All in all, the ratios of typical temperature and entropy of two kinds of phase transitions tend to natural constant in the large dimension limit, which may imply to understand the nature of two different phase transitions.