Big hp enduro racing
 

From what i've seen of exotic rental car track day rentals and driving schools is they leave the cars running in between sessions which makes sense to minimize the heat/cool cycles.
I've been told by alot of the top race shops that evo's in the 600whp range wont last long doing a season of 20 min lapping days, but would keeping the cars running help with durability?

Not really, nobody's put the LeMans budget into making the Evo survive in any kind of enduro environment. Some have tried but High Boost + High HP = ****ty MPG which isn't good for enduro in any combination.

Evo is more of the sprinter car. I wouldn't trust in the Evo around 600 whp in endurance race, for sure.

I would say you can make an Evo run safely in endurance with good cooling, about 350 - 400 whp. that would be pretty reliable. But i ll be honest , i dont think besides Australia/ Germany i heard evo running endurance race.
Dont get high hopes in endurance race with an Evo with High HP. That is a touring car category.

About left running the car between runs, i dont know really if the car cools down to the safe temperature why they need keep them running, unless the weather is really cold.

I can't see how keeping the car running indefinitely will help with durability. When any car comes off the track you want a cool down lap and then maybe a bit more time in the paddock idling before shut off. From going to one of those places in vegas a few years ago, when a car comes in with the previous driver it is more or less time to start getting the next driver ready. At that point in time it would make sense to leave the car running. The cars at that particular one weren't left running between when they drive in and when I got in the car since I had the first session.