Strictly speaking, the thermosphere is actually much warmer than the atmosphere we experience--on the order of 100's or even a 1000 degrees Celsius, if you're measuring by temperature (the average kinetic energy of molecules). However, since particle density is so low, the number of molecules is quite low, and so total heat content of the thermosphere is low. But since particle count is low, conduction and convection are essentially nonexistent, which means cooling needs to rely entirely on radiation, which is much less efficient than other modes at cooling.

In other words, a) background temperature (to the extent it's even meaningful) is much warmer than Earth's surface and b) cooling is much, much more difficult than on Earth.