Thermodynamic topology of black hole solutions in Cotton gravity coupled to nonlinear electrodynamics

In this study, we investigate the thermodynamic topology of black hole solutions in Cotton gravity coupled to nonlinear electrodynamics by incorporating entropy corrections within the framework of different thermodynamic ensembles. We employ Duan’s ϕ-mapping topological current theory as a mathematical tool to study the topological structures associated with these black hole solutions. This approach allows us to compute topological invariants that the black hole’s thermodynamic behaviour is linked to its underlying geometrical and topological features. Our investigation is carried out under two statistical ensembles: the canonical ensemble, where the charge is fixed, and the mixed canonical ensemble, where both the charge and pressure may vary under certain constraints. These ensemble settings provide a broader context for exploring how the thermodynamic properties and topological defects of black holes evolve with changes in entropy, pressure, and other physical parameters. The analysis is further enriched by considering modified entropy models, which reflect to quantum gravitational corrections. By analyzing the behavior of the thermodynamic variables and the corresponding vector fields in each ensemble, we identify the critical points, phase transitions, and topological charges that characterize the system. Our findings demonstrate that topological methods provide powerful insights into black hole thermodynamics, suggesting new ways to interpret stability and phase structure in the presence of conformal Killing gravity and entropy formalisms. This work not only enhances our understanding of black hole thermodynamics, but also provides a novel perspective on how topological classifications can reflect physical phenomena in gravitational systems. It opens up new avenues for exploring black holes in alternative theories of gravity, especially where standard thermodynamic analysis may be insufficient.