Kalen--I still have to think that the ground surface temperature is above freezing. I can't see them letting you play if the air temp is 25 with overcast skies, but I may be wrong there. I would still think if the ground surface is frozen, it would greatly damage the turf.
I've done it several times on several courses including TP and a handful of others in the SLC and Spokane area.
Granted that being said, most courses will just close for the winter time, perhaps for this reason or for others....but many of the SLC valley courses are open all winter long, snow permitting.
A funny story to go along with this....
Some buddies and I were playing a course in the Spokane area. We came to a short par 4 and one of them dubbed it off the tee. He then proceeded to hit a beautiful high towering 5 wood that was all over the pin. It hit the green and took a huge bounce in the air like it had hit concrete. The green was frozen solid with a layer of ice directly on top of it.
Needless to say we all finished the hole, but it took "fast and firm" to a new level. It was kinda fun because none of us could even get a short wedge shot to stay on the green so we had to kind of finess it with a putter until we eventually holed out!!