Thermodynamic topology of AdS black holes with $F^{\alpha \beta }{F}^{\gamma \lambda }{R}_{\alpha \gamma }{R}_{\beta \lambda }$ coupling in the grand canonical ensemble

We investigate the thermodynamic and topological phase structure of charged AdS black holes with a nonminimal gauge–curvature coupling in the grand canonical ensemble. In the minimal-coupling limit $\epsilon =0$, the system exhibits only Hawking–Page-like behavior and does not possess van der Waals criticality. Working perturbatively in the nonminimal coupling, we find that the $\epsilon \mathrm{\ne }0$ interaction generates a rich phase structure as the electric potential is varied, including several distinct regimes of critical behavior, a double-critical region, and a regime where conventional criticality disappears. We complement the conventional thermodynamic analysis with a topological description based on the winding number of the thermodynamic vector field. The resulting $r_h$–$\tau$ diagrams reveal that the ordering and morphology of the black-hole branches depend sensitively on both the pressure and the off-shell inverse temperature, with monotonic, disconnected, S-shaped, and cusp-like structures appearing in different parameter ranges. Remarkably, these substantial rearrangements of the solution branches can occur without changing the global winding number. In particular, the system remains in the $W=+1$ topological class beyond the range of conventional criticality, before eventually entering a $W=0$ sector. Our results demonstrate that thermodynamic topology provides complementary global information that is not captured by conventional local criticality criteria alone.