Corrections to the statistical entropy of five dimensional black holes

We compute the statistical entropy of the three charge (D1-D5-p) five dimensional black hole to sub-leading order in a large charge expansion. We find an agreement with the macroscopic calculation of the Wald entropy in R-2 corrected supergravity theory. The two calculations have a overlapping regime of validity which is not the Cardy regime in the microscopic conformal field theory. We use this result to clarify the 4d-5d lift for black holes on Taub-NUT space. In particular, we compute sub-leading corrections to the formula S-4d = S-5d. In the microscopic analysis, this correction arises from excitations bound to the Taub-NUT space. In the macroscopic picture, the difference is accounted by a mechanism present in a higher derivative theory wherein the geometry of the Taub-NUT space absorbs some of the electric charge.