Abstract
Cavity electrodynamics is emerging as a promising tool to control chemical processes and quantum material properties. In this work we develop a formalism to describe the cavity mediated energy exchange between a material and its electromagnetic environment. We show that coplanar cavities can significantly affect the heat load on the sample if the cavity resonance lies within the frequency region where free-space radiative heat dominates, typically the mid-IR at ambient temperature, while spectral filtering is necessary for having an effect with lower frequency cavities.