XUV (10 nm ⩽ λ ⩽ 100 nm) free-electron lasers (FELs) are potentially important light sources for advanced lithography and materials applications. The average power of an XUV FEL oscillator may be limited by thermal loading of the resonator mirrors. We analyze the requirements for the thermal performance of the mirrors of a metal, multifacet-mirror ring resonator for use at 12 nm. We use analytical methods and numerical approaches which include simulations with the 3-D FEL code FELEX. Thermal distortion of mirror surfaces leads to optical wavefront aberrations which reduce the focusability of the light beam in the gain medium (wiggler/electron beam) and limit the laser performance.