  We study the generalized free energy of the five dimensional charged
Gauss-Bonnet 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 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.
