  We discuss the thermodynamic aspects of a single qubit based device, powered
by weak quantum measurements, and feedback controlled by a quantum Maxwell's
demon. We discuss both discrete and time-continuous operation of the
measurement based device at finite temperature of the reservoir. In the
discrete example where a demon acquires information via discrete weak
measurements, we find that the thermodynamic variables including the heat
exchanged, extractable work, and the entropy produced are completely determined
by an information theoretic measure of the demon's perceived arrow of time. We
also discuss a realistic time-continuous operation of the device where the
feedback is applied after a sequence of weak measurements. In the
time-continuous limit, we derive the exact finite-time statistics of work, heat
and entropy changes along individual quantum trajectories of the quantum
measurement process, and relate them to the demon's arrow of time.
