I don't think that's a problem. The FPR lets fuel slowly back into the tank. As long as it holds the proper pressure while the engine's running it's OK. Just make sure it's not leaking OUT somewhere. You'd probably smell it if it were leaking.

When you turn the key 'on' the fuel pump only runs for about 2 seconds. If the ECM doesn't see a tachometer signal from the distributor by then, it shuts off the fuel pump. I think that's some kind of safety feature in case you're in a collision.