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Using the Bragg-Williams construction of an off-shell free energy, we calculate the topological charge of the Hawking-Page transition point for black holes in anti--de Sitter. A computation following from a related off-shell effective potential in the boundary gauge dual matches the value of topological charge obtained in the bulk. We also work out the topological charges of the equilibrium phases of these systems, which follow from the saddle points of the appropriate free energy. The locally stable and unstable phases turn out to have topological charges opposite to each other, with the total being zero, in agreement with the result obtained from a recent construction S.-W. Wei, Y.-X. Liu, and R. B. Mann, arXiv:2208.01932.
Yerra et al. (Fri,) studied this question.