Testing the AdS/CFT correspondence through thermodynamic geometry of nonlinear electrodynamics AdS black holes with generalized entropies

Abstract We investigate the thermodynamics and thermodynamic geometry of three classes of Anti-de Sitter (AdS) black holes arising from nonlinear electrodynamics, namely ModMax, nonlinear electrodynamics (NED), and Euler–Heisenberg black holes, together with their holographically dual conformal field theory (CFT) descriptions. The analysis is performed using the standard Bekenstein–Hawking entropy as well as the generalized R’enyi and Kaniadakis entropy formalisms. The phase structure is characterized through the temperature, specific heat, and the Legendre-invariant scalar curvature of geometrothermodynamics (GTD). We show that thermodynamic critical points coincide with extrema of the temperature–entropy curve, divergences of the specific heat, and singularities of the GTD curvature, establishing a consistent geometric characterization of phase transitions. A comparison between the bulk black holes and their dual CFTs further demonstrates that the number, ordering, and structure of the critical points are preserved under the holographic correspondence, providing a nontrivial consistency check of bulk-boundary thermodynamics rather than a proof of the AdS/CFT correspondence. The Euler–Heisenberg AdS black hole exhibits the richest phase structure among the models considered, while the Kaniadakis entropy consistently introduces an additional critical point absent in the Bekenstein–Hawking and R’enyi descriptions. These results illustrate how nonlinear electromagnetic interactions and generalized entropy formalisms jointly influence the thermodynamic behavior of AdS black holes and their holographic duals.