Thermodynamic topology of AdS black holes with $F^{αβ}F^{γλ}R_{αγ}R_{βλ}$ 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. Working perturbatively in the nonminimal coupling, we find 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 $r_h$--$τ$ 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.