Thermodynamics and phase transition in the Kerr-de Sitter black hole

It is well known that there is a black hole and cosmological horizons in a Kerr-de Sitter spacetime. The thermodynamic quantities on both horizons satisfy the laws of black hole thermodynamics. In this paper, by considering the relations between the two horizons, we give the effective thermodynamic quantities in axially symmetric de Sitter spacetime. The thermodynamic properties of these effective quantities are analyzed. Moreover, the critical temperature and the critical pressure are obtained. We conclude that the phase transition point is irrelevant to the angular momentum. It only depends on the ratio of the black hole horizon and the cosmological horizon. This result is consistent with the nature of the liquid-gas phase transition at the critical point and, thus, deepens the understanding of the analogy of the rotating de Sitter spacetime and liquid-gas systems.