pretty sure it's always the same rate. First barbarian always seems to spawn at turn #10 on a camp that was there from the start.
Glad you enjoyed it, you should check out how the AIs suffer from it, they really can't handle that kind of aggression
I find it a lot easier to catch up to them on immortal / diety when the AI struggles to stay alive in the early game. It feels like win/win, the early game is far more interesting and you can go up a difficulty level.