  We introduce a general variational framework to address the tunneling of hot
Fermi systems. We use the representation of the trace of the imaginary time
$\tau=it$ propagator as a functional integral type of a sum over complete sets
of states at intermediate propagation slices. We assume that these states are
$\tau$-dependent and generated by an arbitrary trial Hamiltonian $H_0(\tau)$.
We then use the convexity inequality to derive $H_0(\tau)$ controlled
variational bound for a trial action functional. This functional has a general
structure consisting of two parts - statistically weighted quantum
penetrability and dynamical tunneling entropy. We examine how this structure
incorporates the basic physics of tunneling of hot Fermi systems. Using the
variational inequality one can optimise the dynamical parameters controlling
the action functional for any choice of the trial problem.
  As an application we take $H_0(\tau)$ to describe imaginary time dynamics of
non interacting Bogoliubov-de Gennes (BdG) quasiparticles. Optimising its
dynamical parameters we extend the tunneling theory of hot Fermi systems to the
Hartree-Fock-Bogoliubov(HFB) frame and derive the corresponding generalisation
of imaginary time temperature dependent BdG mean field equations.
  As in the trial action the prominent feature of these equations is an
inseparable interplay between quantum dynamical and entropic statistical
effects. In the zero temperature limit these equations describe the "false
ground state" tunneling decay of superfluid Fermi systems (spontaneous fission
in nuclear physics). With increasing excitation energy (effective temperature)
the decay process is gradually evolving from pure quantum tunneling to
statistical "bottle neck" escape mechanism.
