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We study the generalized free energy of the five dimensional chargedGauss-Bonnet AdS black holes in the grand canonical ensemble by treating theblack hole radius and the charge as the order parameters. On the twodimensional free energy landscape, the lowest points in the basins representthe local stable black holes and the saddle point represents the unstable blackhole. We show that black hole is the topological defect of gradient field ofthe landscape. The black hole stability is determined by the topography of thefree energy landscape in terms of the basin depths and the barrier heightbetween the basins and is not by the topology of the gradient field. Inaddition, we study the stochastic dynamics of the black hole phase transitionand obtain the dominant kinetic path for the transition on the free energylandscape. Unlike the one dimensional landscape, the dominant kinetic pathbetween the small and the large black hole state does not necessarily passthrough the intermediate black hole state. Furthermore, the inhomogeneity indiffusions can lead to the switching from the coupled cooperative process ofblack hole phase transition to the decoupled sequential process, givingdifferent kinetic mechanisms.

Topology of the landscape and dominant kinetic path for the thermodynamic phase transition of the charged Gauss-Bonnet AdS black holes