Depends.
If the carcass is cooled quickly outside, with the skin on, it's November (winter), cool room is cool. I have aged Deer and Elk just like beef up to 21 days.
There are plenty of times when this is not possible or of interest and have processed meat the same day. Typically, it won't hang for more than 7 days.
Great article on the subject that was the result of research done at the University of Wyoming.
"Hunters seldom agree as to the length of time a big game carcass should be aged.
What is involved in the aging process? When is it beneficial to age game meat? Under what conditions is it inadvisable to age game? This pamphlet is concerned with answers to these questions.
Let's assume that the hunter has made his kill and properly dressed the carcass. Now we want to know what he should do from the time the carcass is eviscerated until it is ready to be cut into steaks and roasts."
Summary
"Many practical considerations must ultimately determine whether to age or not to age game meat. Among these are the temperature at the time of kill, the chilling rate, the internal temperature of the muscle after chilling, the youthfulness of the animal, the relative humidity, the amount of weight loss the hunter is willing to sacrifice, the processing procedure and the cooler space and labor available if the game is to be processed commercially. Under ideal conditions, age antelope 3 days, deer, sheep, goat, cow elk and cow moose 7 days and bull elk and bull moose 14 days after the kill at 34ºF. If the temperature is higher, the aging period should be shorter. Game which is killed when the temperature is 65ºF or above and held at this temperature over 1 day should be cut immediately. Game that is to be ground or chopped does not need to be aged."
Aging Big Game
— by Ray A. Field and C. Colin Kaltenback
Bulletin 513R, Agricultural Experiment Station, University of Wyoming, Laramie, WY 82071. Publication date: October 1987.