We studied the effects of cryogenic cooling of a 2-[3-(4-hydroxystyryl)-5, 5-dimethylcyclohex-2-enylidene] malononitrile (OH1) crystal on the generation of broadband THz pulses via collinear optical rectification of 1350 nm femtosecond laser pulses. Cooling of the OH1 crystal from room temperature to 10 K leads to a ~10% increase of the pump-to-THz energy conversion efficiency and a shift of the THz pulse spectra to a higher frequency range. Both effects are due to the temperature variation of the THz absorption and the refractive index of the OH1 crystal. This conclusion has been verified by temperature dependent measurements of the linear absorption in the THz frequency region. An approach to obtain a stronger increase of the THz generation efficiency at cryogenic cooling of the OH1 crystal is discussed.