  BPS black hole degeneracies can be expressed in terms of an inverse Laplace
transform of a partition function based on a mixed electric/magnetic ensemble,
which involves a non-trivial integration measure. This measure has been
evaluated for black holes with various degrees of supersymmetry and for N=4
supersymmetric black holes all results agree. It generally receives
contributions from non-holomorphic corrections. An explicit evaluation of these
corrections in the context of the effective action of the FHSV model reveals
that these are related to, but quantitatively different from, the
non-holomorphic corrections to the topological string, indicating that the
relation between the twisted partition functions of the latter and the
effective action is more subtle than has so far been envisaged. The effective
action result leads to a duality invariant BPS free energy and arguments are
presented for the existence of consistent non-holomorphic deformations of
special geometry that can account for these effects. A prediction is given for
the measure based on semiclassical arguments for a class of N=2 black holes.
Furthermore an attempt is made to confront some of the results of this paper to
a recent proposal for the microstate degeneracies of the STU model.
