Therin,
I don't know how I developed this idiosyncracy, but I hold my bow in hand ALL the time while hunting. It's not as convenient as my buddy's way of hanging the bow until a deer comes into sight, but it keeps you ready for the "only 10 seconds from sight to shot" drill. Two of the deer I killed this year wouldn't have been possible if I'd had to reach for a bow because of how and where they came in.
The coldest I've been out this year is 10 degrees, which is surely no minus 7, but I've done really well by keeping my core warm and slipping a footwarmer pouch under the back of my fleece glove on my bow hand.
I've seen three benefits. I'm ready.... I can use the bow as a mini "blind" between me and the approaching deer (it's amazing how it seems to help break up the outline)... and I "practice draw" from time to time to keep warm and keep confidence high.
Drawing hand is in my pocket if it's cold, but bow is in hand across my lap. Males me feel like I'm ready and "on duty!"
It's all worth it when the snow turns red!!
Good luck in your hunts!
Daryl