Thermodynamics and orbital structure of anti-de Sitter black holes in Palatini-inspired nonlinear electrodynamics

Abstract We construct a consistent anti-de Sitter completion of the static and spherically symmetric black-hole solution sourced by the Palatini-inspired nonlinear electrodynamics $$Y^n$$ Y n model. Here, “Palatini-inspired” denotes exclusively a first-order formulation of the electromagnetic sector, in which the gauge potential and an auxiliary antisymmetric tensor are varied independently; the gravitational sector is the standard metric Einstein–Hilbert theory with the Levi-Civita connection. Starting from the Einstein–Hilbert action with a negative cosmological constant and the first-order PINLED sector, we derive the full set of field equations and show that the nonlinear electromagnetic solution preserves its original parametric structure, while the lapse function acquires the standard AdS contribution. We then analyze the horizon structure, Hawking temperature, extended phase-space thermodynamics, and the associated equation of state. In addition, we investigate null and timelike geodesics, with emphasis on the effective potentials, photon sphere, critical-curve (shadow boundary) radius for a static observer at finite distance, and innermost stable circular orbit. The resulting framework furnishes the exact AdS extension of the asymptotically flat PINLED black hole and provides a coherent basis for numerical and phenomenological studies of its thermodynamic, optical, and orbital properties.