  If two phases exist at the same time, such as a gas and a liquid, they have
the same temperature. This fundamental law of equilibrium physics is known to
apply even to many non-equilibrium systems. However, recently, there has been
much attention in the finding that inertial self-propelled particles like Janus
colloids in a plasma or microflyers could self-organize into a hot gas-like
phase that coexists with a colder liquid-like phase. Here, we show that a
kinetic temperature difference across coexisting phases can occur even in
equilibrium systems when adding generic (overdamped) self-propelled particles.
In particular, we consider mixtures of overdamped active and inertial passive
Brownian particles and show that when they phase separate into a dense and a
dilute phase, both phases have different kinetic temperatures. Surprisingly, we
find that the dense phase (liquid) cannot only be colder but also hotter than
the dilute phase (gas). This effect hinges on correlated motions where active
particles collectively push and heat up passive ones primarily within the dense
phase. Our results answer the fundamental question if a non-equilibrium gas can
be colder than a coexisting liquid and create a route to equip matter with
self-organized domains of different kinetic temperatures.
