Here is one magic number for you 600F - that is about the chamber temperature where you will begin to see cook-offs because the chamber is hot enough to detonate the primer. You can have cook-offs at much lower temperatures than that, particularly if the ammo has been stored badly and the primers have become unstable; so don't think you are safe just because you are at 550F.
On ARs, I wouldn't sweat it too much. You can hit barrel temps of 300F pretty regularly and your standard chrome-lined barrel will still be good for hitting silhouettes out to 600yds for about 10k rounds. You may start losing consistency at longer ranges around 5k though.
On AKs, I wouldn't sweat it either as there isn't a whole lot of precision to use.
With standard handguards on either rifle, a good rule of thumb is to stop shooting and let it cool when you see smoke coming off the handguards (you can ignore white smoke from AR handguards, that is just CLP burning off - black smoke is the glue holding the metal heat shields burning and that is a good sign to stop).
If you want the absolute maximum possible life out of your barrel, just stop shooting whenever the barrel gets too hot to hold with your bare hand; but you can exceed that standard quite a bit and get and still get 10-20k rounds of use out of a barrel.