Due to the high intensity and MHz repetition rate of photon pulses generated by the European X-ray Free-Electron Laser, the heat load on silicon crystal monochromators can become large and prevent ideal transmission in Bragg diffraction geometry due to crystal deformation. Here, we present experimental data illustrating how heat load affects the performance of a cryogenically cooled monochromator under such conditions. The measurements are in good agreement with a depth-uniform model of X-ray dynamical diffraction taking beam absorption and heat deformation of the crystals into account.