I have this minor problem with my Mazda 6 - sometimes it doesn't start up because fuel pump is not engaging. Usually this happens after running a short trip (15minutes or so) and stopping the engine after that. I have checked fuel pump's fuse and switched the fuel pump's relay too. It seems that fuel pump is not getting power and the pump itself is working since it starts when it's connected directly to the power supply.

Any ideas what to check next?

 

Otherwise car cranks up normally, just doesn't start up.

 

Common issue with these older 6's is the fuel pump dying a slow death. Double check for power & earth to the pump to be sure.

My money is on the pump.

 

There was bad contact on the base of the fuel relay box.