Not familiar with the electrics of this car but here goes.

I gather you are saying that
1) The starter motor turns and engine starts when cold
2) The starter motor does NOT turn when warm

If the starter has been replaced then you need to look to see if the Starter
Solenoid Switch is integral with the starter motor or is a separate
switch/relay.

Follow the heavy duty wire back from the starter all the way to the battery.
If the lead goes directly to the battery without going through any other
device then the starter solenoid must be attached to/integral with the
starter motor.

Goes without saying that all connections should be properly checked.  Check
for tightness and corrosion.  Also check the ignition switch feed for the
started solenoid supply.

Nick /////