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 $$F^{\alpha \beta }F^{\gamma \lambda }R_{\alpha \gamma \beta \lambda }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>F</mml:mi> <mml:mrow> <mml:mi>α</mml:mi> <mml:mi>β</mml:mi> </mml:mrow> </mml:msup> <mml:msup> <mml:mi>F</mml:mi> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mi>λ</mml:mi> </mml:mrow> </mml:msup> <mml:msub> <mml:mi>R</mml:mi> <mml:mrow> <mml:mi>α</mml:mi> <mml:mi>γ</mml:mi> <mml:mi>β</mml:mi> <mml:mi>λ</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> . 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 $$W^{1+}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> </mml:math> and the grand canonical ensemble within class $$W^{0-}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>-</mml:mo> </mml:mrow> </mml:msup> </mml:math> . Our results demonstrate that the non-minimal coupling preserves the universal topological classification while allowing ensemble choice to fundamentally alter the phase structure.