sounds like me
I went through a very similar problem myself.
1) Is the starter working? Meaning, do you just here a whirring sound of the starter trying to engage, or do you hear the engine trying to turn over?
2) Are the fuel pumps working? Turn the key without trying to start the car. Have someone crouch by the back and listen for the fuel pumps turning on when the key is turned. You'll a brief high-pitched buzz of some sorts if they are.
3)Is the fuel pump relay ok? On my 420 SEL is was on the driver's side, by the firewall. Abou the size of a pack of cigarettes. Honestly, not sure what you should look for when you pull it out. I believe there is some type of indiactor on the top that tells you if it's blown or not. On mine, it's a white dot. Again, not 100% on what to look for, but it could be a source of the problem.
Here's what my mechanic told me. When the car was cold it started fine becasue the relay was cool in temperature. After I had run the car for a bit, the relay had a chance to heat up beyond a temperature that would allow it work, so the car wouldn't start. That's why the car would probably start after the few hours.
Hopefully, you'll be able to find out that it's only the relay since this is a much cheaper option.
You could also be having problems getting a good spark, so check the plugs and ignition coil.
Hope this helps.