Engine shuts off after 12 minutes

So, before I get in to it I'll cover what was done along the way.  All parts verified to be operating correctly.  Replaced fuel pump, fuel pressure regulator, fuel relay.
The car ran fine and then one day decided to do what it's doing.

When the engine is cold you can start it and it will do everything it should do.  Rather that is sitting idling, or driving it down the road.
Always at between 12-15 minutes since the time it was started it will turn off.  When it does it will not start again until the engine has cooled off, or enough time has passed.  Rather it is engine temp that is a factor or just time at rest I have not been able to nail down.  The car is not overheating, or close to it.  Not even to the point that it would kick on the aux fans.
When it shuts down it is the fuel pump stopping.  If you spray some fuel in the intake it will start and run as long as it's getting fuel.  I checked the issue and found the pump is turning off.  If you by-pass the hot wire to the pump the ECU will then kill the ignition.  Of course without spark it's doing nothing.

What is telling/making the ECU(it's my guess this is what's doing it)to turn the fuel pump power off?  Is there a sensor that can be bad that triggers this response?  It's a '97, but it's still a beautiful car.  Give me a set of points and a carb any day.