Why does my gravity vented wall Furnace keep turning off/ [solved]

We have an Empire gravity vented wall furnace:

That turns off. According to the instructions, this is because the vent safety switch. But when I open up the furnace, it appears to be vented just fine, in fact, I can feel the cool air pouring down.

After resetting the switch, it will work until the heat goes off (when the room reaches temperature) but then will not turn back on unless I reset it again, and that is erratic too.

So my question is is there anything I can do to make the safety switch work again? I should probably call a service person, but was curious if anyone has specific experience with these types of wall furnaces.