  We consider the AdS_2/CFT_1 holographic correspondence near the horizon of
rotating five-dimensional black holes preserving four supersymmetries in N=2
supergravity. The bulk partition function is given by a functional integral
over string fields in AdS_2 and is related to the quantum entropy via the Sen's
proposal. Under certain assumptions we use the idea of equivariant localization
to non-rigid backgrounds and show that the path integral of off-shell
supergravity on the near horizon background, which is a circle fibration over
AdS_2xS^2, reduces to a finite dimensional integral over n_V+1 parameters C^A,
where n_V is the number of vector multiplets of the theory while the C^0 mode
corresponds to a normalizable fluctuation of the metric. The localization
solutions, which rely only on off-shell supersymmetry, become after a field
redefinition, the solutions found for localization of supergravity on
AdS_2xS^2. We compute the renormalized action on the localization locus and
show that, in the absence of higher derivative corrections, it agrees with the
four dimensional counterpart computed on AdS_2x S^2. These results together
with possible one-loop contributions can be used to establish an exact
connection between five and four dimensional quantum entropies.
