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We study the generalized free energy of the five dimensional charged Gauss-Bonnet anti--de Sitter (AdS) black holes in the grand canonical ensemble by treating the black hole radius and the charge as the order parameters. On the two dimensional free energy landscape, the lowest points in the basins represent the local stable black holes and the saddle point represents the unstable black hole. We show that the black hole is the topological defect of gradient field of the landscape. The black hole stability is determined by the topography of the free energy landscape in terms of the basin depths and the barrier height between the basins and is not by the topology of the gradient field. In addition, we study the stochastic dynamics of the black hole phase transition and obtain the dominant kinetic path for the transition on the free energy landscape. Unlike the one dimensional landscape, the dominant kinetic path between the small and the large black hole state does not necessarily pass through the intermediate black hole state. Furthermore, the inhomogeneity in diffusions can lead to the switching from the coupled cooperative process of black hole phase transition to the decoupled sequential process, giving different kinetic mechanisms.
Li et al. (Thu,) studied this question.