in a sterile desert, more life will grow if the temperature falls

No. In a sterile desert more life will grow if there is more water available at the surface. Of course - more water available at the surface will also cause surface temperatures to fall, due to increased evaporative cooling. Temperature is an issue in many hot arid areas - such as the Sahara - due to lack of water. The equator gets more sunlight above it, but also has more clouds, and more water (at the surface) to cool it. So water is the issue in arid areas - not temperature.

BTW: In some arid regions of China day-time, growing season, temperatures fell by over 6C when mass irrigation was introduced. See: Yang/Huang/Tang, 2019