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.

Technically radiation cooling is 100% efficient. And remarkably effective, you can cool an inert object to the temperature of the CMBR (4K) without doing anything at all. However it is rather slow and works best if there's no nearby planets or stars.

Fun fact though, make your radiator hotter and you can dump just as much if not more energy then you would typically via convective cooling. At 1400C (just below the melting point of steel) you can shed 450kW of heat per square meter, all you need is a really fancy heat pump!