This one is a strange one. Every now and then after I turn off the car I can hear the fuel pump keep on spinning. I noticed that when I open the gas cap and let the air hiss...the fuel pump stops spinning. What gives?

I have been told that it's normal with air hiss/swoosh sound when opening the gas cap, but now I'm not sure.

Not the best mechanic on earth, but this one has truly left me totally clueless

 

Check your fuel pump relay.

 

When you say check your fuel pump relay, do you mean to check the contact points or to test it. If so how do you go about testing the relay?

Come to think about it, I have pulled the relay a couple of times, open the gas cap and then crank the ignition in order to depressurize the fuel system.
I know this is not the procedures according to the manual, but the only way I have learned to do it. Could this procedure actually have damaged the fuel system?

 

Does anyone know if the fuel pump relay is suposed to activate the pump with the ignition in the on position (without cranking engine) or does it start the pump only by cranking and running the engine?

I have searched around and some people state that on a 1978 280z the engine needs to run.

On my car the pump will start spinning with the ignition in the on position...and keep on spinning...never shuts off. On "normal" cars the pump should stop spinning after approx. 5 sec. with the ignition on.

Sometimes after a drive the pump will keep on spinning even after I shut the engine off. As soon as I open the gas cap (hissing air), the pump stops spinning. I have another tread on this problem, and NismoPick refers to the fuel pump relay.

Feedback is very much appreciated

 

Time for you to download the factory service manual. The 78 is wired through the oil sender so that if the engine stalls the pump shuts off. The fuel filter might be clogged up causing te pump to not reach full pressure to cycle on/off. If the fuel pressure regulator is bad it won't let the system get to the 35 psi required. There should be a filter before the pump and one under the hood. Sounds like the pump is starving which explains your "hiss" diagnosis.