Swampland motivations and extremality diagnostics for nonminimally coupled magnetic black holes

We investigate the thermodynamic behavior and extremal configurations of magnetically charged black holes in nonminimal Einstein-Yang-Mills theory, focusing on the interplay between nonminimal interactions, local stability, and thermodynamic topology. The derived mass and Hawking temperature recover the Reissner-Nordström-AdS results in the minimally coupled limit. However, integrating the first law leaves an entropy-normalization ambiguity, so conclusions based on entropy positivity remain conditional on the prescription adopted here. Our stability analysis distinguishes the extremal boundary from divergences in the heat capacity, which indicate singular local responses but do not independently establish a first-order phase transition. The topological analysis further shows that the winding number depends on both magnetic charge and pressure. Restricting the thermodynamic domain to positive entropy can alter the endpoint contribution, making a separate evaluation necessary. Regarding extremality, the asymptotically flat solution admits a positive extremal branch throughout the considered nonnegative coupling range, while the AdS case possesses a unique positive extremal radius under the stated conditions. The extremal mass increases with the nonminimal coupling, and its response remains finite at finite positive coupling but grows without bound in the large-coupling limit. These findings clarify the geometric and thermodynamic consequences of the nonminimal interaction without overstating their implications for quantum gravity. In particular, the extremality results alone neither establish nor rule out the magnetic Weak Gravity Conjecture, whose assessment requires a consistently normalized gauge sector and information about the spectrum of charged states.