We discuss the power dependence of distributed cavity phase errors for cylindrical TE011 cavities in laser-cooled atomic fountain clocks. The azimuthally symmetric phase variations produce a surprisingly large distributed cavity phase error for two 2π, 4π, and 6π pulses. This is due to the correlation between the transverse variation of the Rabi frequency over the cavity aperture and a quadratic density variation of the atomic sample, along with the symmetry of the longitudinal phase variation in the cavity. We show that the large azimuthally symmetric fields and phase shifts near the walls of the endcap holes produce very small errors at optimal power for a uniform wall resistance. We also show the power variation for higher order azimuthal variations m=1, 2, and 4. These may be caused by fountain tilts, non-uniform detection of atoms, and asymmetries in the laser trapping and cooling of the atoms. We demonstrate that distributed cavity phase errors in physical cavities may have no variation with the microwave power. A combination of rigorous calculations of cavity losses, measurements of power dependence, the atomic distributions, and fountain tilts, and electrical measurements that show the lower limit of the cavity Q and the cavity symmetry, should provide stringent limits on distributed cavity phase errors for current atomic clocks.