  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.
