I would check the fuel pressure first in a case like this. Or I would bypass the fuel pump resistor to see the difference. My guess is that the fuel pump is loosing its capability to send enough fuel to the fuel rail when heated up. An aged fuel pump tends to show that type of characteristic.
The fuel pump is cooled down by the fuel, but when the fuel flow is not enough, the generated heat is not dissipated well and the torque of the motor is dropped resulting less fuel to be sent. Too much accumulated heat triggers the Posistor to stop the motor to run for the safety and there comes no fuel to the rail.
ECU capacitors are almost the due for aged cars especially at warmer places, but your current problem is a bit different from the capacitor issue. Aged capacitors tend to show problems when cold, I must add.