Exponential Corrected Metric of 3D BTZ Black Hole

We investigate the thermodynamics of the (2+1)-dimensional BTZ black hole in the presence of a non-perturbative exponential correction to the Bekenstein–Hawking entropy. Adopting an effective entropy deformation within a horizon thermodynamics framework, we reconstruct the corresponding corrected metric and derive the modified temperature, heat capacity, and free energies. The correction is exponentially suppressed for large black holes, ensuring recovery of the classical AdS$_3$ solution, while producing controlled deviations at small horizon radii. We investigate the thermodynamics of the (2+1)-dimensional BTZ black hole in the presence of a non-perturbative exponential correction to the Bekenstein–Hawking entropy. Adopting an effective entropy deformation within a horizon thermodynamics framework, we reconstruct the corresponding corrected metric and derive the modified temperature, heat capacity, and free energies. The correction is exponentially suppressed for large black holes, ensuring recovery of the classical AdS$_3$ solution, while producing controlled deviations at small horizon radii. We find that exponential corrections generate a nontrivial intermediate thermodynamic regime and alter stability properties relative to the classical BTZ case. Our results provide a simple analytic model for exploring non-perturbative quantum effects in lower-dimensional gravity.