  We investigate higher derivative corrections to the extremal Kerr black hole
in the context of heterotic string theory with $\alpha'$ 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 $\alpha'^2$. 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.
