  We extend the definition of non-adiabatic entropy production given for
Markovian systems in [M. Esposito and C. Van den Broeck, Phys. Rev. Lett. 104
090601, (2010)], to arbitrary non-Markov ergodic dynamics. We also introduce a
notion of stability characterizing non-Markovianity. For stable non-Markovian
systems, the non-adiabatic entropy production satisfies an integral fluctuation
theorem, leading to the second law of thermodynamics for transitions between
non-equilibrium steady-states. This quantity can also be written as a sum of
products of generalized fluxes and forces, thus being suitable for
thermodynamics. On the other hand, the generalized fluctuation-dissipation
relation also holds, clarifying that the conditions for it to be satisfied are
ergodicity and stability instead of Markovianity. We show that in spite of
being counter-intuitive, the stability criterion introduced in this work may be
violated in non-Markovian systems even if they are ergodic, leading to a
violation of the fluctuation theorem and the generalized
fluctuation-dissipation relation. Stability represents then a necessary
condition for the above properties to hold and explains why the generalized
fluctuation-dissipation relation has remained elusive in the study of
non-Markov systems exhibiting non-equilibrium steady-states.
