I would have to say it is a bad sensor. In a few minutes it should not be too hot to trip the sensor. I'm not sure what temperature they are set to trip at if you can take it off and look at it there should be numbers stamped on it.
As a very temporary check, unhook the sensor and connect the terminals together and see if it will stay lit. If it stays on longer than it was, you have isolated the problem I would think.