Chen-Hao Hao, Jieci Wang

We develop a branch-sensitive thermodynamic framework for higher-curvature black holes using the off-shell Gibbs free energy $G_{\rm off}$ and the Wald entropy$S_W$ as the basic data. On fixed-parameter slices, equilibrium black holes are stationary points of $G_{\rm off}$, and their local stability is governed by the Hessian $H=S'_W(r_h)T'(r_h)$, rather than by the temperature slope alone. For the five-dimensional charged regular AdS black hole in quasi-topological gravity, $S_W$ remains monotonic on the physical branch, so the usual temperature-slope rule is recovered only as a special consequence. The same off-shell structure also gives the local $A_3$ cusp normal form near criticality, yielding the mean-field $1/2$ branch separation exponent and explaining why smooth nondegenerate observables, such as the Lyapunov exponent, inherit the same scaling. In Lovelock black holes, $S'_W$ can change sign on non-planar branches, reversing the temperature slope stability assignment. However, on ghost-free and branch-regular Lovelock exteriors $S'_W$ remains positive. Thus the off-shell Hessian criterion also diagnoses why the ordinary slope rule is protected on physically admissible black holes branches.