I have always wondered if cooling it hanging with the gut wedged open versus skinning it immediately was preferable. My gutted deer are generally at the processer within 5 hours max and I didn't think this an issue even with fairly warm temps (50's and 60's).
All I can do is repeat that In a commercial kitchen I have 4 hours to bring the temp of meat to below 45 Degrees F.
I know that this isn't possible to do that in the field most of the time. Still it is a very good idea to get that carcass cold asap.
One thing that people might consider is wiping or rinsing out the carcass with a strong vinegar solution.