After starting the car you have hold excelarator padel down at 2000 revs until it warms. if dont hold padel down it dies.

Year make and model, and have you checked base fuel pressure with a gauge ?

hello ther I belive it is your ecu very common

Your toyota technician should know what a "cold starting solenoid" is. It is the little thing that works your throttle until the motor warms up and something else takes over. I'd suggest going to another technician.

cold idle control sensor bad

Sounds like a coolant sensor problem. It signals the ECU for open loop, cold start rich, high idle, until warm up is reached.

Check under hood for a vacuum hose off,broken, cracked or loose at a connector.

try to replace the fuel pump

linked to your accelerator padel is a switch which tell the ecu how much speed the engine should gain when you press the padel. If its faulty then it doesn't come back or comes slowly when you move your foot from padel and as a result it malfunctions. spray a little turpentine oil or lamp oil on this switch will resolve your problem.

check for vacuum leak at the intake area,also check EGR vale may be stuck open.