Hawking--Page phase transitions of the black holes in different extended phase spaces

The Hawking–Page (HP) phase transitions of the d-dimensional Schwarzschild and charged black holes are explored in two different extended phase spaces. One is to enclose the black hole in the anti-de Sitter (AdS) space, and the other is to confine the black hole in a spherical cavity. The phase transition temperature THP, minimum black hole temperature T0, and Gibbs free energy G are systematically calculated in an analytical way. There are remarkable similarities in the two extended phase spaces. Especially, for the Schwarzschild black holes, a dual relation of THP(d) = T0(d + 1) in successive dimensions exactly valid in the AdS case is found to be also approximately valid in the cavity case with a high precision. Moreover, this relation can be further generalized to the charged black holes in a suitable form. At the same time, significant dissimilarities also exist between the two extended phase spaces, like the notable terminal points in the G–T curves of the charged black holes in a cavity. Our work helps to establish the universal properties of the black holes in different extended phase spaces, and simultaneously motivates further studies on their thermodynamic behaviors that are sensitive to specific boundary conditions. Corresponding author. ar X iv :2 10 5. 06 67 0v 1 [ gr -q c] 1 4 M ay 2 02 1