Always starts cold or after sitting 30 minutes. Cranks but doesn't start 2nd or 3rd time after quick, short stops.

 

Could be a failing fuel pump.

If having the right tools can check fuel pressure. An OBDII scanner with Toyota code reading capability might show up something.

Replacing the pump is a DIY job. One can get the complete housing assy which would be a remove and replace the task or just the pump which would need to be installed on the housing assy. Not a big deal but more effort..

 

Issue has been ongoing for several months, wouldn't fuel pump have failed by now. Seems to be related to engine temperature somehow since waiting awhile after driving allows starting success.

 

Maybe yes, maybe no. Could be the fuel pump has become weak. The fuel pressure regulator connected located at the pump could have problems plus there is a check valve to prevent pressure bleed down.

Have someone try to start the car hot while you sniff the tailpipe for signs of gas odor. Could be a leaking injector causing a flooding condition. Do NOT depress the gas pedal when starting engine as the engine should start fine without doing this..

The engine computer has a sensor to tell it coolant temp. The sensor could be having problems and sending the wrong signal to the computer. Here an ODBII scanner could read the sensor temp.