  We apply the monodromy method for the calculation of the functional
determinant of a special second order differential operator
$F=-d^2/d\tau^2+{\ddot g}/g$, $\ddot g= d^2g/d\tau^2$, subject to periodic
boundary conditions with a periodic zero mode $g=g(\tau)$. This operator arises
in applications of the early Universe theory and, in particular, determines the
one-loop statistical sum for the microcanonical ensemble in cosmology generated
by a conformal field theory (CFT). This ensemble realizes the concept of
cosmological initial conditions by generalizing the notion of the no-boundary
wavefunction of the Universe to the level of a special quasi-thermal state
which is dominated by instantons with an oscillating scale factor of their
Euclidean Friedmann-Robertson-Walker metric. These oscillations result in the
multi-node nature of the zero mode $g(\tau)$ of $F$, which is gauged out from
its reduced functional determinant by the method of the Faddeev-Popov gauge
fixing procedure. The calculation is done for a general case of multiple nodes
(roots) within the period range of the Euclidean time $\tau$, thus generalizing
the previously known result for the single-node case of one oscillation of the
cosmological scale factor. The functional determinant of $F$ expresses in terms
of the monodromy of its basis function, which is obtained in quadratures as a
sum of contributions of time segments connecting neighboring pairs of the zero
mode roots within the period range.
