The time-dependent variational principle proposed by Balian and Vénéroni is used to provide the best approximation to the generating functional for multi-time Green’s functions of a set of (bosonic) observables Qµ. By suitably restricting the trial spaces, the computation of the two-time Green’s function, obtained by a second order expansion in the sources, is considerably simplified. This leads to a tractable formalism suited to quantum fields out of equilibrium. We propose an illustration on the finite temperature Φ4-theory in curved space and coupled to gravity.