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

Abstract

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--Nordstr\"om--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.