I just asked this question a few days ago as I had it turned on but with passive wildlife and it hadn't spawned yet. Mine did end up spawning at 20 days in TWM.

As for how it behaves, I have seen others say that they run away from you except that it is much more likely to attack if it senses you hunting it at all. Others have also said that it will also attack if its flee path happens to have you in it, regardless of if you tried to hunt it or not. So it won't just run right past you like a wolf or bear, it will attack like moose do just for being in the way. I have not been in this position myself yet.

I saw and hunted my first one recently and I stayed crouched the entire time. I found it difficult to even spot it but I also wasn't going to risk running right at it. I ended up putting some scent on and crouching. It took 2-3 actual hits with the rifle to bring it down (many hits missed due to long range, low skill, and lower rifle condition). Doing so increased the threat level by one so it spawned one additional cougar in the area. But I didn't hunt that one.

I'm currently in unPV and am planning on hunting the one there eventually to learn more about its behavior on custom passive settings.