  The established thermodynamic formalism of chaotic dynamics, valid at
statistical equilibrium, is here generalized to systems out of equilibrium,
that have yet to relax to a steady state. A relation between information,
escape rate, and the phase-space average of an integrated observable (e.g.
Lyapunov exponent, diffusion coefficient) is obtained for finite time. Most
notably, the thermodynamic treatment may predict the phase-space profile of any
finite-time integrated observable from the leading and subleading
eigenfunctions of the Perron-Frobenius/Koopman transfer operator. Examples of
that equivalence are shown, and the theory is tested analytically on the
Bernoulli map, while numerically on the perturbed cat map, the H\'enon map, and
the Ikeda map, all paradigms of chaos.
