The response of silicon drift detectors (SDDs), which were mounted together with their preamplifiers inside a vacuum chamber, was studied in a temperature range from 100 K to 200 K. In particular, the energy resolution could be stabilized to about 150 eV at 6 keV between 130 K and 200 K, while the time resolution shows a temperature dependence of T^3 in this temperature range. To keep a variation of the X-ray peak positions within 1 eV, it is necessary to operate the preamplifier within a stability of 1 K around 280 K. A detailed investigation of this temperature influences on SDDs and preamplifiers is presented.