Do bow companies not sell bows in the places like Arizona because the 120 degree weather might make the explode? Or do they attach warnings?
Dyneema (the most heat intolerant material I'm aware of being used in bow strings) is rated for prolonged exposure up to 70 C (158 F) and short excisions up to 100 C (212 F). There are places in the US where the ambient temperature in a car might get above 158 F, but probably not the case for most of us.
158 F will result in 3rd degree burns in a few seconds with hot liquids. Granted a solid surface will contain less heat energy and transfer it less effectively, but it is still a temp you will not be wanting to touch. Other that directly sun exposed materials, even on 115 degree days in San Antonio, TX things never got close to that hot in a vehicle to my memory.
So, definitely keep it out of the sun, a cooler is probably overkill, but if it makes you feel better and does not add too much hassle, who cares.