  Coupling a system to two different baths can lead to novel phenomena escaping
the constraints of thermal equilibrium. In quantum materials inside optical
cavities, this feature can be exploited as electrons and cavity-photons are
easily pulled away from their mutual equilibrium, even in the steady state.
This offers new routes for a non-invasive control of material properties and
functionalities. We show that the absence of thermal equilibrium between
electrons and photons leads to reduced symmetries of the steady-state
electronic distribution function. Moreover, by defining an effective
temperature from the on-shell distribution function, we find a non-monotonic
behaviour as a function of cavity frequency, consistent with recent
experimental findings. Finally, we show that, the non-thermal behaviour leads
to qualitative modifications of the material's properties, as the standard
Sommerfeld expansion for observables is modified by a leading-order correction
linearly proportional to the temperature difference between the two baths and
to the frequency-derivative of the electron damping.
