I took the panel off and saw that it's controlled by a 15amp 120V thermoswitch. I'm not really sure exactly at what temp it comes on but it does come on when the house is around 90*F so I'm guessing that's what it's open/close temp.
I found one that looks very similar but is adjustable:
https://www.grainger.com/product/GRA...n-Switch-6UEE1
I think if I adjust this one to open/close at 70* then it will be closer than what it is now and work much more efficiently at 40*F to 60*F. But first I have to make sure it will fit correctly.
An Arduino that could compare two temperatures and then close a relay would be better of course since I could chose 70*F or more room temp whenever I want without adjusting the fan switch temp.