We study the leading $\alpha \prime$ corrections to the entropy of certain black holes with AdS$5\times S5$ asymptotics. We find that, in the supersymmetric limit, the entropy does not receive $\alpha \prime$ corrections. This result strengthens recent calculations that match the index of $N=4$ Super-Yang-Mills with the corresponding partition function in the supersymmetric limit. In the small temperature regime, we find that the entropy corrections are concordant with the weak gravity conjecture.