  We study a class of non-equilibrium quasi-stationary states for a Markov
system interacting with two different thermal baths. We show that the work done
under a slow, external change of parameters admits a potential, i.e., the free
energy. Three conditions are needed for the existence of free energy in this
non-equilibrium system: time-scale separation between variables of the system,
partial controllability (external fields couple only with the slow variable),
and an effective detailed balance. These conditions are facilitated in the
continuous limit for the slow variable. In contrast to its equilibrium
counterpart, the non-equilibrium free energy can increase with temperature. One
example of this is that entropy reduction by means of external fields (cooling)
can be easier (in the sense of the work cost) if it starts from a higher
temperature.
