On the dual relation in the Hawking--Page phase transition of the black holes in a cavity

The Hawking--Page phase transitions of the $d$-dimensional Schwarzschild and charged black holes are explored in a cavity. The phase transition temperature $T_{\rm HP}$, the minimum black hole temperature $T_0$, and the Gibbs free energy $G$ are systematically calculated. A dual relation for the Schwarzschild black holes in the anti-de Sitter space, $T_{\rm HP}(d)=T_0(d+1)$, is found to be also approximately valid in the cavity case to a high precision, and this relation can be further generalized to the charged black holes in a suitable form. Our work reveals the universal properties of the black holes in different extended phase spaces and motivates further studies on their thermodynamic behaviors that are sensitive to specific boundary conditions, like the terminal points in the $G$--$T$ curves.