I believe you are looking for a reverse acting thermostat. We used to have one that was mounted near the top of a vaulted ceiling in a room with a wood stove. It would activate a fan that circulated air from the top of the room to registers throughout the house. Not really any different from what any thermostat does in cooling mode in a conventional HVAC system.
There are circulating fans sold in the stores that sell wood stoves. They are activated by a bimetal switch that makes contact when the unit gets hot. You probably want a bimetal air temperature switch. Not sure where you could buy one, but probably at a wood stove store or good electrical / plumbing supply store. The stack gets hot very fast and using the stack to trigger the fan would probably start circulation air prematurely before the stove gets hot. It would also be a difficult place to mount switch wiring. I recommend that you mount the temperature sensor on a wall behind or above the stove where it detects heat after the wood stove gets warm enough to warm the space around it. This way you could use a standard air conditioning thermostat to activate a fan when the temperature goes above say 80 degrees in the area around the stove. That way you will be circulating warm or hot air.