  The equation of motion for the two-fermion two-time correlation function in
the pairing channel is considered at finite temperature. Within the Matsubara
formalism, the Dyson-type Bethe-Salpeter equation (Dyson-BSE) with the
frequency-dependent interaction kernel is obtained. Similarly to the case of
zero temperature, it is decomposed into the static and dynamical components,
where the former is given by the contraction of the bare interaction with the
two-fermion density and the latter is represented by the double contraction of
the four-fermion two-time correlation function, or propagator, with two
interaction matrix elements. The dynamical kernel with the four-body
propagator, being formally exact, requires approximations to avoid generating
prohibitively complicated hierarchy of equations. We focus on the approximation
where the dynamical interaction kernel is truncated on the level of two-body
correlation functions, neglecting the irreducible three-body and higher-rank
correlations. Such a truncation leads to the dynamical kernel with the coupling
between correlated fermionic pairs, which can be interpreted as emergent
bosonic quasibound states, or phonons, of normal and superfluid nature. The
latter ones are, thus, the mediators of the dynamical superfluid pairing. In
this framework, we obtained the closed system of equations for the fermionic
particle-hole and particle-particle propagators. This allows us to study the
temperature dependence of the pairing gap beyond the Bardeen-Cooper-Schrieffer
approximation, that is implemented for medium-heavy nuclear systems. The cases
of 68Ni and 44,46Ca are discussed in detail.
