I know my PT425 will heat up enough just sitting still at 1/3 idle to warm up the hydraulics after half an hour to the point the cooler fan will come on. That's just from the fluid going through the two PTO pumps and back to the tank. However, I wouldn't be concerned too much. There's got to be more of a load swinging a mower in thick grass AND driving VS just sitting and running a generator... I'd imagine you'd have to get a pretty hefty electric load on it before it starts putting a load on the PTO pump and hydraulics.
It would be interesting to put a load on the generator and monitor the temps of the generator head, motor, PTO pumps, hydraulic reservoir, cooler, etc...