Gutting in the field does two things with respect to cooling:
1. Immediately removes a lot heat that is in the guts.
2. Provides far more surface area for cooling. And cooling is all about surface area - see car radiator or AC fins as examples.

So you are speeding up the cooling process at perhaps more than double the rate. Without gutting, the deer can only cool through the hide, which is made to help keep the deer warm in sub-zero weather.