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Quantum Statistical Relation for black holes in nonlinear electrodynamics coupled to Einstein-Gauss-Bonnet AdS gravity

We consider curvature-squared corrections to Einstein-Hilbert gravity action in the form of Gauss-Bonnet term in D>4 dimensions. In this theory, we study the thermodynamics of charged static black holes with anti-de Sitter (AdS) asymptotics, and whose electric field is described by nonlinear electrodynamics (NED). These objects have received considerable attention in recent literature on gravity/gauge dualities. It is well-known that, within the framework of anti de-Sitter/Conformal Field Theory (AdS/CFT) correspondence, there exists a nonvanishing Casimir contribution to the internal energy of the system, manifested as the vacuum energy for global AdS spacetime in odd dimensions. Because of this reason, we derive a Quantum Statistical Relation directly from the Euclidean action and not from the integration of the First Law of thermodynamics. To this end, we employ a background-independent regularization scheme which consists in the addition to the bulk action of counterterms that depend on both extrinsic and intrinsic curvatures of the boundary (Kounterterm series). This procedure results in a consistent inclusion of the vacuum energy and chemical potential in the thermodynamic description for Einstein-Gauss-Bonnet AdS gravity regardless the explicit form of the NED Lagrangian.