All,
I've had this problem several times, where there wasn't sufficient cooldown post-flight and it wouldn't start on the next attempt. Trying to start it wouldn't get above about 800 rpm and would then quit.
The fix we did was to get access to the turbo and spin the blades and move the axle of it a bit to make sure it was fully free spinning. That always worked and the engine would start normally.
But, today, the plane was taxied to the pumps (by my partners), and a cool down wasn't done because it barely got above idle (~3000 rpm). And it didn't start, just like before. But, this time, the spin-the-turbo trick did not work.
New Additional Info: Just questioning my partners and they say it wasn't warmed up to 120F before they started taxiing (and the RPM was briefly at 3000 rpm).... What damage can that do and how would I fix it?