From quasinormal modes to logarithmic corrections: Role of geometric deformation in dilaton black hole thermodynamics

We investigate the quasinormal mode (QNM) spectrum of a four–dimensional dilaton black hole with a deformed angular geometry [Formula: see text], where [Formula: see text] and [Formula: see text] and its connection to logarithmic corrections in black hole entropy. By analyzing scalar perturbations in the background spacetime, we derive the effective potential for the Schrödinger-like wave equation and compute the complex QNM frequencies using the third-order WKB approximation. The parameter N significantly influences the shape of the potential and the damping rate of perturbations, leading to distinct dynamical behaviors for generic N, [Formula: see text] and [Formula: see text]. We propose a conjectural relation between the asymptotic QNM frequencies and the logarithmic correction coefficient c in [Formula: see text], given by [Formula: see text], which is supported by WKB analysis and numerical fits and reduces to the known value [Formula: see text] when [Formula: see text]. For [Formula: see text] and [Formula: see text] (which lie in the physical range [Formula: see text] but outside the domain of validity of the asymptotic formula), the system exhibits critical or unstable behavior. These results demonstrate that the geometric deformation parameter N acts as a fundamental control parameter in the phase structure and quantum thermodynamics of dilaton black holes. We further verify that in the limit [Formula: see text], [Formula: see text], the solution reduces to the Reissner–Nordström–AdS black hole, confirming the model as a consistent deformation of classical black holes.