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Designs for future gravitational wave detection facilities feature silicon test masses at cryogenic temperatures to reduce thermal noise and thermally induced aberrations. Designers call for operation at 123 K or close to 18 K to exploit the vanishing thermal expansion of crystalline silicon. The amount of absorbed heat that can be radiatively removed from the test masses is limited at these temperatures, forcing complex cooling scenarios to be considered, including conduction through suspension wires. This is particularly relevant for the kilohertz class of detectors that aim for extremely high circulating power, i.e., roughly a factor 20 more than the world record at the time of writing, to reduce quantum noise. We explore the impact of raising the test mass temperature and show that a dedicated kilohertz-band cryogenic instrument can do so without significant sensitivity penalty, thereby boosting the radiative cooling rate and allowing higher power operation with simpler suspensions. We also explore the implications of operating cryogenic broadband detectors at elevated temperatures. The work presented here was instrumental in the development of the Neutron Star Extreme Matter Observatory kilohertz-band gravitational wave detector design concept.