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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 $τ=it$ propagator as a functional integral type of a sum over complete sets of states at intermediate propagation slices. We assume that these states are $τ$-dependent and generated by an arbitrary trial Hamiltonian $H_0(τ)$. We then use the convexity inequality to derive $H_0(τ)$ 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 $H_0(τ)$ 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.