Thermodynamic topology of 4D charged AdS black holes with $$F^{\alpha \beta }F^{\gamma \lambda }R_{\alpha \gamma \beta \lambda }$$ coupling

Abstract We investigate the thermodynamic phase transitions of a four-dimensional charged anti-de Sitter black hole endowed with a non-minimal coupling of the form $$F^{\alpha \beta }F^{\gamma \lambda }R_{\alpha \gamma \beta \lambda }$$ F α β F γ λ R α γ β λ . Using perturbative methods, we derive a consistent black hole solution and analyze its thermodynamics through both conventional equilibrium techniques and a topological defect classification approach. In the canonical ensemble, the system displays van der Waals-like critical behavior, with a swallow-tail structure in the free energy and distinct phase branches. In the grand canonical ensemble, by contrast, the system exhibits a Hawking–Page-like transition, characterized by a single divergence in the heat capacity and the absence of a swallowtail. The topological analysis independently confirms both types of critical behavior: it classifies the canonical ensemble within class $$W^{1+}$$ W 1 + and the grand canonical ensemble within class $$W^{0-}$$ W 0 - . Our results demonstrate that the non-minimal coupling preserves the universal topological classification while allowing ensemble choice to fundamentally alter the phase structure.