As high-average power ultrafast lasers become increasingly available for nonlinear conversion, the temperature dependence of the material properties of nonlinear crystals becomes increasingly relevant. Here, we present temperature-dependent THz complex refractive index measurements of the organic crystal BNA over a wide range of temperatures from 300 K down to 80 K for THz frequencies up to 4 THz for the first time. Our measurements show that whereas the temperature-dependent refractive index has only minor deviation from room temperature values, the temperature-dependent absorption coefficient decreases at low temperature (-24% from 300 K to 80 K). We additionally compare these measurements with conversion efficiency and spectra observed during THz generation experiments using the same crystal actively cooled in the same temperature range, using an ultrafast Yb-laser for excitation. Surprisingly, the damage threshold of the material does not improve significantly upon active cooling, pointing to a nonlinear absorption mechanism being responsible for damage. However, we observe a significant increase in THz yield (+23%) at lower temperatures, which is most likely due to the reduced THz absorption. These first findings will be useful for future designs of high-average power pumped organic-crystal based THz-TDS systems.