My 2004 V8 grand marquis wont start when hot. If i allow the engine to cool it starts immediately starts. I suspect its the fuel pump.

 

I think you should start with the Fuel filter first.If that won't cure the issue try checking what you have for fuel pressure. You say after it warms up the engine will not start correct, So the fast way to see if your fuel pump is working is to get into the car ,close the doors and turn the key to make the dash lights come on. At the same time see if you hear the fuel pump start the stop after turning the key on.Don't try to start the car just turn the key to make the dash lights come on and make sure your doors are closed because the chime from the door being open makes hearing the fuel pump for that few seconds hard to hear. If all that fails and you do hear the fuel pump you might want to check to see if the fuel regulator is working.I don't know what you have in the means of Tools so I would just say you would have to hook up a Scan Tool which is able to record the data of the engine while driving and also reads any stored codes that are in the memory of the on-board CPU Also known as your cars brain. Auto zone or Advanced Auto will hook up a scanner to your car most of all at least where I live for free.I found that buying my own was a good investment because you need to learn a lot of things about how newer cars before just replacing parts because a code comes up while someone scans your car. Most of the time the scan tool will tell you what system of the engine by means of a code where to start looking for the problem , then it's up to you to find the correct issue. Please let me know your mileage and feel free to contact me if you can't find the problem. Please try not to get ripped off by someone that claims your fuel pump is bad without checking to see if you can hear it running when you turn that key,It could be a simple as a fuel filter and most Mechanics know how to test mostly everything without taking the part of the car.Just a fuel pressure test while running can give some good news or bad. Good Luck. John