  In nano-scale systems coupled to finite-size reservoirs, the reservoir
temperature may fluctuate due to heat exchange between the system and the
reservoirs. To date, a stochastic thermodynamic analysis of heat, work and
entropy production in such systems is however missing. Here we fill this gap by
analyzing a single-level quantum dot tunnel coupled to a finite-size electronic
reservoir. The system dynamics is described by a Markovian master equation,
depending on the fluctuating temperature of the reservoir. Based on a
fluctuation theorem, we identify the appropriate entropy production that
results in a thermodynamically consistent statistical description. We
illustrate our results by analyzing the work production for a finite-size
reservoir Szilard engine.
