Gut shot deer in warm weather??
When a deer is hit and there is minimal trauma ( no big bones impated, no gun going off etc) they will normally run a bit and duck into cover, from which they can watch their back trail. If you stay in your tree and quiet, or sneak out the back and leave them, they will likely bed right there and you will later find them, right there.
As stated, if you follow too soon, the deer will run again and you may not find it.
Congrat's on finding it. ChuckC
I would gamble if it was me. Give it 6 hours and go for it. You'll either find it and the meat will be okay or it might get up and run. At least then you know the meat is okay yet I guess. But if you wait too long the whole thing is shot, equivalent of losing it completely.