I've come to find the fan is most active between 80 and 100 degrees ambient. Living in Phoenix, that's most of the year, so our P300 fan spins up a lot after shutting the engine off. Funny enough when it's 100+ it doesn't do it as much, which I suspect is because the fan is already on high while the engine is running. Anything below that, and the fans aren't usually running at max speed while the engine is on.

It's actually impressive how much heat it removes from the engine bay in the 20-30 seconds that it runs. I'd imagine that helps with the longevity of some components under the hood.

 

Fan is super loud when just driving at small speed at parking lot, if stopped right after highway it's not that loud and runs for less.
There's is circulation pump that runs for few minutes pretty much always and it looks like it cools turbo.
I think overall P300 runs somewhat hot, it is subjective opinion of course,

 

In the old days ? I'm so old I still do that That's also why I don't use the Stop/Start feature ................call me old fashioned, luddite, stick in the mud - I don't care; at least I'm not cooking the oil in the turbo every few minutes.