  This paper is concerned with the stochastic thermodynamics of non-equilibrium
Gaussian processes that can exhibit anomalous diffusion. In the systems
considered, the noise correlation function is not necessarily related to
friction. Thus, there is no conventional fluctuation-dissipation relation (FDR)
of the second kind and no unique way to define a temperature. We start from a
Markovian process with time-dependent diffusivity (an example being scaled
Brownian motion). It turns out that standard stochastic thermodynamic notions
can be applied rather straightforwardly by introducing a time-dependent
temperature, yielding the integral fluctuation relation. We then proceed to our
focal system, that is, a particle undergoing fractional Brownian motion (FBM).
In this system, the noise is Gaussian but the noise correlation function is
nonlocal in time, defining a non- Markovian process. We analyse in detail the
consequences when using the conventional notions of stochastic thermodynamics
with a constant medium temperature. In particular, the heat calculated from
dissipation into the medium differs from the log ratio of path probabilities of
forward and backward motion, yielding a deviation from the standard integral
fluctuation relation for the total entropy production if the latter is defined
via system entropy and heat exchange. These apparent inconsistencies can be
circumvented by formally defining a time-nonlocal temperature that fulfils a
generalized FDR. To shed light on the rather abstract quantities resulting from
the latter approach we perform a perturbation expansion. This allows us to
calculate analytically, up to linear order, the generalized temperature and the
corresponding heat exchange. By this, we provide explicit expressions and a
physical interpretation for the leading corrections induced by
non-Markovianity.
