We investigate higher derivative corrections to the extremal Kerr black hole
in the context of heterotic string theory with α′ corrections and of a
cubic-curvature extension of general relativity. By analyzing the near-horizon
extremal geometry of these black holes, we are able to compute the Iyer-Wald
entropy as well as the angular momentum via generalized Komar integrals. In the
case of the stringy corrections, we obtain the physically relevant relation
S(J) at order α′2. On the other hand, the cubic theories, which are
chosen as Einsteinian cubic gravity plus a new odd-parity density with
analogous features, possess special integrability properties that enable us to
obtain exact results in the higher-derivative couplings. This allows us to find
the relation S(J) at arbitrary orders in the couplings and even to study it
in a non-perturbative way. We also extend our analysis to the case of the
extremal Kerr-(A)dS black hole.Comment: 33 pages+appendices, one figur