  The path integration technique recently developed for the statistical sum of
the microcanonical ensemble in cosmology is applied to the calculation of the
one-loop preexponential factor in the cosmological model generated by a
conformal field theory with a large number of quantum species -- the model of
initial conditions possibly related to the resolution of the cosmological
constant and landscape problems. The result is obtained for the family of
background cosmological instantons with one oscillation of the FRW scale
factor. The magnitude of the prefactor is analytically and numerically
estimated for fields of various spins conformally coupled to gravity, which
justifies the validity of semiclassical expansion for this family of
cosmological instantons.
