It doesn't have a thermostat. I believe the pump runs when the engine is running.
Forum tradition requires at least three people to say the same thing for it to be true.
While Gary is correct, the "supercharger pump" is mechanically driven by a belt off the crankshaft pulley. Thus when the engine is running (the crankshaft is spinning) the supercharger pump is in operation by default.
Now perhaps Tipo874 is referring to another system loosely associated with the supercharging circuit. Important to use and possibly understand the correct terminology to extract the answer you are looking for.
The 5.4 engine is fed compressed inlet air by a mechanical supercharger discussed above but before the compressed air is allowed to enter the cylinder for combustion, it is cooled by an "intercooler". This heat exchanger sits immediately below the supercharger, cools the compressed discharge air for better engine volumetric efficiency, has its own separate cooling circuit, a circuit dedicated "radiator" in the front of the vehicle and the pump moving the fluid in the intercooler circuit is thermostatically controlled.
I do not know of the top of my head the on/off pump activation limits for the intercooler circuit (if that is the information you seek). Perhaps Rich (GT Guys) might know.
