Yes i have2002 saturnsl 1.9 i have replace fuel pump,fuel filter,temp. sensor it will run until the heat gauge reaches 1/4 and it just dies when cools back down starts up does it again.

by not running the fault codes first , you have missed the real faulty sensor that is causing the problem
nothing of what you replaced is linked to hard starting, like , ambient temp sensor ( used to report the air temp for the setting of the rich mixture need for starting)
baro sensor ( used for reporting the air density used to set the mixture setting
Maf/ Vaf sensors ( again used for mixture settings)
heated oxygen sensors ( used to adjust the gas composition by reporting the results to the ECM

Had a 89 cavalier did that start then die it map sensor. Also check ground wires if the become too corroded they won't start sometimes just run weird

The cold start system is not keeping the engine running. There is fuel to fire up but then the temp sensor that keeps it running until water or manifold temp is up is not working. Look at the temp sensors.

try checking the wiring for broken or worn though .use a ohms meter to see if power wires earth out etc.

It could be something with the EGR valve, or the O2 sensor, or fuel pressure.
I would probably check fuel pressure cold and hot first.

you might have problems with the coils
check the temp of the coils when it stops working
if faulty they heat up alot

possibly 4 things 1 bad ignition coil heating up then breaking down,2 clogged fuel filter,3 weak fuel pump,4 bad throttle position sensor or map sensor to check sensors simply put on hand held scanner and look for codes. replace fuel filter replace coil pack along with tune up plugs wires put fuel pressure gauge to see if fuel pump is working properly

Its normal for the pump to run for three seconds then shut off if the car engine is off. . This indicates that the fuel pump turned on and the pressure did reach specifications. If the pump did not reach pressure the pump would continue to run, You need to check for proper pressure with a gauge. If tthe pressure is ok then you have another issue like an ignition module or crank sensor if applicable. You may have to drive the car with the fuel gauge hooked up to see if you still have pressure when it dies.. I know you said you replaced the pump already but do you have the same issues as before you replaced it. This would indicate which direction you need to go now. If the same then its prabably not the pump.

Does the fuel line still have 55 PSI when engine warmed up? Any other codes show up?