Entropy-Driven Reorganization of Thermodynamic Topology in AdS–Rindler Black Holes

We study the thermodynamics and thermodynamic topology of AdS–Rindler black holes with non-perturbative exponential entropy corrections. Using the proper hyperbolic horizon area, we obtain a first-law-consistent semiclassical thermodynamics in which the heat capacity has a single zero but no finite divergence. The exponential correction S η = S + η e − δ S preserves this classical zero, but can generate new extrema of the corrected temperature and hence genuine heat-capacity singularities absent in the uncorrected system. By determining the critical correction strengths η c from the stationary points of η * ( r + ) , we show that the first thresholds lie slightly above unity for d = 3 , 4 , 5 . Consequently, weak corrections η ≤ 1 only shift the large-horizon stable branch, whereas strong corrections η = 3 , 5 reorganize the near-horizon sector. In the off-shell thermodynamic-topology framework, black-hole branches appear as zero points of a vector field in the ( r + , Θ ) plane, with winding number w = sign ( d T η / d r + ) . For η ≤ 1, each dimension contains a single stable defect with total charge W = 1 . For η = 3 , 5 , the admissible phase space develops a stable–unstable defect pair with net charge W = 0 , separated from the excluded region by the singular radius r s = ( ln η ) 1 / ( d − 1 ) . Thus, strong exponential entropy corrections do not merely deform AdS–Rindler semi classical thermodynamics; they drive a thermodynamic-topological reorganization of the entropy-deformed fixed-background phase space.