  A quantum system and a thermal bath can reach thermal equilibrium through an
interaction, whereupon the system acquires the same temperature as the bath.
But how does a delocalised quantum system thermalise with a bath whose local
temperature varies, as, for example, in the Tolman effect? Here we formulate
two scenarios in which the notion of a ``superposition of temperatures'' may
arise. First: a probe interacting with two different baths dependent on the
state of another quantum system (control). Second: a probe interacting with a
single bath whose purified state is a superposition of states corresponding to
different temperatures. We show that the two scenarios are fundamentally
different and can be operationally distinguished. Moreover, we show that the
probe does not in general thermalise even when the involved temperatures are
equal, and that the final probe state is sensitive to the specific realisation
of the thermalising channels. Our models may be applied to scenarios involving
joint quantum, gravitational, and thermodynamic phenomena, and explain some
recent results found in quantum intereference of relativistic probes
thermalising with Unruh or Hawking radiation. Finally, we show that our results
are reproduced in partial and pre-thermalisation processes, and thus our
approach and conclusions hold beyond the idealised scenarios, where
thermalisation is incomplete.
