Cooling a board is tricky if the entire evironment is hot and there's no supply of cooling air. You can cool parts locally to some extent using a Peltier heat pump, but they generate heat of their own which has to be managed somehow.
Most electronics can be located somewhere more benign - so, for example, if you're monitoring and controlling an internal combustion engine, the ECU doesn't have to be in the engine bay. The sensors themselves do, though, so if you've got something producing a small signal that needs to be locally measured, digitised and sent up a cable, then the hardware to do that needs to tolerate whatever the temperature happens to be in the place where the signal is available.
Down hole electronics do certainly fall into this area too, and there's certainly no ready supply of cooling air half a mile underground. It's a bit of a niche industry, though. Not too many customers... though those which do exist can afford expensive kit.
Are you licensing an ARM core and making a high temperature chip yourselves, or building modules with other peoples' chips?