The VP44 is a rotary pump. The pump has to rotate to put fuel through it. It is not possible to get fuel through it with zero rotation (shutdown).
IF that were possible then that would be the best cooling (use of fluid vs air, lubrication, etc), but not possible. That is why we went to external air flood cooling. The air cooling also allows air to go through the APPS pocket (if you drill a couple of holes in the top of the APPS pocket) to help cool the APPS as well (after all it is an electronic component). Bob Weis