Logarithmic corrections to the entropy of extremal black holes have proven effective in precisely matching the microscopic degeneracies obtained from string-theoretic as well as a non-perturbative quantum correction manifests as an exponential term in the black hole entropy. In this work, we extend the universal relation proposed by Goon and Penco by deriving a generalized form where entropy is not just the Bekenstein-Hawking entropy. Our analysis treats entropy as a general function of the horizon radius, and with the help of that, we formulate the generalized universal relation. We show that, in the case of Bekenstein-Hawking entropy, the generalized relation coincides with the original universal relation by Goon and Penco. Furthermore, we explore the implications of logarithmic and exponential corrections to entropy and test the validity of the generalized universal relation under these modifications.
Ankit Anand (Tue,) studied this question.