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 [Formula: see text]. First, by calculating the HP temperature [Formula: see text], the minimum temperature [Formula: see text] and the Gibbs free energy [Formula: see text] of the AdS black hole in different values of parameter [Formula: see text], we find that the HP phase transition can occur at all pressures. Comparing the HP temperature [Formula: see text] with the minimum temperature [Formula: see text], we notice that the ratio of [Formula: see text] tends to 1 when [Formula: see text]. 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 [Formula: see text], the small–large black hole phase transition occurs at the same time. Furthermore, the black hole entropy also exists [Formula: see text] when [Formula: see text], and [Formula: see text] when the black hole temperature has its minimum [Formula: see text], we surprisingly find that the ratio of [Formula: see text] tends to [Formula: see text] when the dimension [Formula: see text], 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.