A simple theory is developed for calculating the contribution from thermal diffuse scattering to the absorption of high energy electrons passing through crystals. The Einstein model for a vibrating crystal is used and the incident wave is assumed to be a Bloch wave, the thermally diffuse scattered wave to be a plane wave. It is shown that thermal diffuse scattering makes an important contribution to absorption (in agreement with a treatment due to Yoshioka & Kainuma 1962), particularly for elements of high atomic number. The theory also predicts a strong anomalous absorption effect for low order reflexions. The mean and anomalous absorption coefficients are strongly temperature dependent. An improvement of several times the penetration at room temperature is predicted for the penetration at low temperatures, for elements of high atomic number and low Debye temperature $\theta$. In practice the improvement realized is probably limited by another, less temperature-dependent, absorption process due to electronic excitations. Careful experimental measurements of absorption coefficients at low temperatures are needed.