  We analyse near-horizon solutions and compare the results for the black hole
entropy of five-dimensional spherically symmetric extremal black holes when the
N=2 SUGRA actions are supplied with two different types of higher-order
corrections: (1) supersymmetric completion of gravitational Chern-Simons term,
and (2) Gauss-Bonnet term. We show that for large BPS black holes lowest order
\alpha' corrections to the entropy are the same, but for non-BPS are generally
different. We pay special attention to the class of prepotentials connected
with K3\times T^2 and T^6 compactifications. For supersymmetric correction we
find beside BPS also a set of non-BPS solutions. In the particular case of T^6
compactification (equivalent to the heterotic string on $T^4\times S^1$) we
find the (almost) complete set of solutions (with exception of some non-BPS
small black holes), and show that entropy of small black holes is different
from statistical entropy obtained by counting of microstates of heterotic
string theory. We also find complete set of solutions for K3\times T^2 and T^6
case when correction is given by Gauss-Bonnet term. Contrary to
four-dimensional case, obtained entropy is different from the one with
supersymmetric correction. We show that in Gauss-Bonnet case entropy of small
``BPS'' black holes agrees with microscopic entropy in the known cases.
