Thermodynamic and Topological Phase Transitions of AdS Black Holes with Nonminimal $F^{\alpha \beta }{F}^{\gamma \lambda }{R}_{\alpha \gamma }{R}_{\beta \lambda }$ Coupling

The conventional and topological phase transitions of a four-dimensional asymptotically AdS black hole with a non-minimal coupling term $F^{\alpha \beta }{F}^{\gamma \lambda }{R}_{\alpha \gamma }{R}_{\beta \lambda }$ are investigated. Using a perturbative approach to first order in the coupling $\epsilon$, the thermodynamic quantities are derived and the first law and Smarr relation are verified. Intriguingly, while the $\epsilon =0$ limit yields the standard Reissner–Nordström–AdS black hole belonging to the $W^{1+}$ topological class ($W=+1$), switching on the non-minimal coupling fundamentally transforms the topology to the $W^{0-}$ class ($W=0$). This transition occurs while the van der Waals-type first-order phase transition survives in the intermediate region, embedded within the overall Hawking–Page pattern, rendering the system a hybrid black hole thermodynamic system. The coupling $\epsilon$ thus acts as a topological deformation parameter that alters the universal classification of the system, despite the perturbative nature of the solution.