Thermal Fluctuations and the Corrected Thermodynamics of Euler–Heisenberg Anti–de Sitter Black Holes

Black holes are bona fide thermodynamic systems, and the leading quantum correction to their entropy is a universal logarithm whose coefficient encodes the microscopic degrees of freedom. We ask how this correction reshapes the thermodynamics of the four-dimensional Euler–Heisenberg anti–de Sitter (EH–AdS) black hole, a charged AdS geometry whose quantum-electrodynamic non-linearity is tuned by a single parameter [Formula: see text], and where it can be trusted. In the extended phase space, with the cosmological constant as pressure and the mass as enthalpy, a microcanonical saddle-point evaluation of the density of states gives [Formula: see text], with [Formula: see text] a model-dependent constant of steepest-descent value [Formula: see text]. Within a leading-order thermal-fluctuation prescription that holds the temperature and geometry at equilibrium and corrects only the statistical entropy, the same logarithm propagates to the internal energy, the free energies, the enthalpy, the equation of state, and the heat capacity, each reducing to its classical, Reissner–Nordström–AdS, and Schwarzschild–AdS limit. Three results stand out within this prescription: the internal energy sheds its cosmological-constant term through a clean cancellation; the Helmholtz and Gibbs free energies share the single correction [Formula: see text]; and the equation of state is uncorrected, so the [Formula: see text]–[Formula: see text] criticality and phase classification stay classical. The heat capacity keeps its two [Formula: see text]-independent Davies divergences and tends to [Formula: see text] at the cold, near-extremal endpoint of evaporation, where a positive [Formula: see text] marginally stabilizes the smallest holes—though exactly where the large-entropy expansion is weakest, a limit we make explicit.