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.
