The performance of a NaI(Tl) crystal coupled to a Large Area Avalanche Photodiode (LAAPD) was investigated as a function of temperature in the range from -40 degC to +23 degC. The number of electron-hole pairs created per 1 MeV of gamma-ray energy detected by the scintillator, non-proportionality and the energy resolution for various peaking times were measured. It showed a significant variation of the light output and the energy resolution of a NaI(Tl) crystal at reduced temperatures down to -40 degC, as measured with different peaking times in the spectroscopy amplifiers. A large slowing down of the light pulse, caused by the intense slow component of the light pulse, requires long peaking times (up to 48 mus) in order to optimize the energy resolution at the most reduced temperatures. In these conditions of a full integration of the light, a weak variation of the light output of a NaI(Tl) crystal is observed, corresponding to -0.16%/degC. The non-proportionality characteristics and intrinsic resolution are somewhat distorted by a partial integration of the light by too short shaping in the amplifier. It confirms the earlier observations that, for crystals which exhibited two component decays of the light pulse, the best non-proportionality and the lowest intrinsic resolution are achieved for the full integration of the light.