Analytic expressions for quasinormal modes of the general parametrized spherically symmetric black holes and the Hod's proposal

Using an expansion in terms of the inverse multipole number and the WKB approach, we derive an analytic expression for the quasinormal frequencies of generic parametrized spherically symmetric and asymptotically flat black holes described by the Rezzolla-Zhidenko spacetime, in the regime when deviations from the Schwarzschild geometry are relatively small, i.e., when the coefficients of the parametrization are small. As an application of this analytic formula, we demonstrate that such generic black holes satisfy Hod's proposal, which constrains the damping rate of the least damped mode in relation to the Hawking temperature.