There are many factors involved and temperature is a key factor, but shot placement also matters. It is not uncommon for daytime highs to be in the 80s during bow season in VA. On occasion, I've paunch shot a deer in the afternoon on a hot day. Rule of thumb is to wait at least 4 hours to trail a paunch shot deer and often longer. I've recovered deer at 9 or 10 o'clock the next morning. Rigor has set in by that time. I've then field dressed, skinned, and quartered the deer.
Spoilage is caused by bacteria multiplying. Heat and moisture create conditions where they can multiply much more quickly and taint the meat. A paunch shot typically exposes the meat to bacteria from the gut. I discard any meat in the area of the shot that was exposed to bacteria.
The rest of the meat from deer like this were fine. I typically won't hang a deer like this in a cooler for long. I try to butcher as soon as possible.
The faster we reduce the temperature the better. I find skinning is much easier on a warm deer. For most of the deer I shoot they are quartered and hanging in the cooler in a few hours. The cooler keeps the temperature 34-36 degrees. This slows bacteria multiplication while allowing time for the natural enzymes in the meat to breakdown muscle tissue and tenderize the meat. I typically hang deer for 1 to 2 weeks before butchering.
Best of luck!