  The path integral is calculated for the statistical sum of the microcanonical
ensemble in a generic time-parametrization invariant gravitational model with
the Friedman-Robertson-Walker (FRW) metric. This represents the first example
of a systematic calculation of the Faddeev-Popov gauge-fixed path integral in
the minisuperspace sector of quantum cosmology. The gauge fixing procedure,
together with gauging out local diffeomorphisms, also handles the residual
symmetries associated with the conformal Killing vector of the FRW metric and
incorporates the Batalin-Vilkovisky quantization technique for gauge theories
with linearly dependent generators. For a subset of saddle-point instantons,
characterized by a single oscillation of the FRW scale factor, this technique
is designed to obtain the one-loop statistical sum in the recently suggested
model of cosmological initial conditions generated by a conformal field theory
with a large number of quantum species.
