I haven't killed the number of deer as many folks on here, so here goes.

Most of our shots are within 75yards. Most of our deer are recovered within 30 mins of the shot. The deer are usually still very flexible. Last year my dad shot a good 3-4yr old big 9 point. We got to him in ~20 mins, and he was pretty stiff already. Still very warm, but his limbs were very stiff. He only ran ~40yards, but it was in rut, and he was following Does.

Is there a reason that he was stiffing up, or just happened?

Ambient temperature is a big factor. Also if he had been running does the build up of lactic acid in the muscles may have contributed, but time and temperature are the main factor in rigor variations (at least in humans).

I suspect the lactic acid because with rigor in humans, as the muscle tissue begins to break down after death, the rigor goes away.

It was November GA late morning cold. 50's.

It's the lactic acid. I've killed several deer that were being run by dogs for a long period of time that were stiff within 30 min.

Muscle contractions are dependent on ATP, which is the body's major energy source. In order for a muscle contraction to release, you have to give it new ATP. So when something dies it is no longer making ATP, so it can no longer relax = rigor mortis.

I'd say in the case of the deer in rut, he had been chasing and his energy stores were already pretty low to begin with so it didn't take him long to stiffen up.

Since he was running does, he was probably "stiff" before you shot him.

Rigamortis will set in quickly on a deer that has been running than it will on one that is relaxed. Shot a doe a few years back that had been chashed by a coyote. I didn't know it until after I shot her. Last day of the season and she had came in fast and stopped when I grunted loudly. I shot her and then the coyote ran off.
She was stiff before I could get her on my cart in about 30 minutes.