We investigate power scaling of ultrashort-pulse enhancement cavities. We propose a model for the sensitivity of a cavity design to thermal deformations of the mirrors due to the high circulating powers. Using this model and optimized cavity mirrors, we demonstrate 400 kW of average power with 250 fs pulses and 670 kW with 10 ps pulses at a central wavelength of 1040 nm and a repetition rate of 250 MHz. These results represent an average power improvement of one order of magnitude compared to state-of-the-art systems with similar pulse durations and will thus benefit numerous applications such as the further scaling of tabletop sources of hard x rays (via Thomson scattering of relativistic electrons) and of soft x rays (via high harmonic generation).