  We construct spinning black hole solutions in five dimensions that take into
account the mixed gauge-gravitational Chern-Simons term and its supersymmetric
completion. The resulting entropy formula is discussed from several points of
view. We include a Taub-NUT base space in order to test recent conjectures
relating 5D black holes to 4D black holes and the topological string. Our
explicit results show that certain charge shifts have to be taken into account
for these relations to hold. We also compute corrections to the entropy of
black rings in terms of near horizon data.
