Of course, I don’t know about YOUR particular machine..but mine operates like this:
The hyd pump output is directly proportional to the engine speed because the pump is directly bolted to the flywheel. The amount of fluid directed to the hydraulic-drive-motor is regulated by the speed lever/pedal (because it controls the swash-plate and therefore the pump piston-travel)... and any excess flow not utilized by the drive-motor bypasses it and passes thru the oil cooler before dumping back into the reservoir.
If low engine speed is used then less “excess” fluid is sent thru the cooler because more fluid is required by the drive motor for similar speeds.
That's not exactly how it works on the larger HST's with a cooler. The oil flow that propels the machine is in a closed loop, meaning the oil flows from the pump to the motor and back to the pump to be sent back to the motor again and again. During this process you will have leakage past the valve plate, pistons and at the slipper feet. This oil leaks into the HST case.
You have a charge pump which supplies makeup oil to replace this oil that has leaked from the closed loop. The charge pump oil is introduced to the low pressure or return side of the closed loop. The charge pump depending on the system also supplies servo oil to actuate the servo which controls the swash plate. Whatever oil from the charge pump that is not used by these functions goes out the charge pressure relief valve and into the HST case.
Basically all the oil that is pumped by the charge pump is sent through the HST case in one path or another and out the case drain port to the cooler and back to the transmission case.
As stated the smaller units do not have a cooler and rely on air flow or the housing for cooling. That is why it is very important to make sure the housing is clean.
Also hydraulic systems should be at a minimum of 160F operating temperature. Normal operating temperature should be about 180F up to 200F.