Topological classification of a 4 <i>D</i> ads black hole with non-minimal maxwell coupling

Abstract We perform a topological classification of the phase structure of a four-dimensional AdS black hole with non-minimal Maxwell coupling. Critical points are treated as topological defects, allowing us to assign a winding number to each black hole branch and compute the global topological invariant W . The system exhibits a duality governed by its Maxwell charge Q : for large Q it falls into the class W = + 1 , displaying van der Waals–type behavior with a first-order small/large black hole transition. For small Q , it shifts to W = 0, characteristic of a Hawking–Page transition. This topological classification provides a model-independent validation of the conventional thermodynamic analysis. Crucially, we find that the non-minimal coupling λ stabilizes the Hawking–Page universality class ( W = 0) for black holes with non-zero charge, a phenomenon absent in the standard Reissner–Nordstr–m–AdS case. This establishes a direct link between the microscopic coupling and the macroscopic topological class, demonstrating the power of topological methods in decoding thermodynamic universality across modified gravity theories.