Current drops affect relay performance , bad connections, open up when hot or warm we see it all the time.
Check your cranking voltage hot and cold, put a volt meter on your battery,and crank int over watch the meter ,if it drops below 10 your going to have problems with efi system components working period. Especially cold starting, the cold start sytem has to have enough voltage to open the cold start injector or it will crank and crank and crank and make the battery voltage worse.
We learned this early on while trouble shooting efi component problems.
The Circuit opening relay is used while cranking and runing as seen in the drawing the Air Flow Meter door opens when cranking making contact with the fuel pump switch which grounds the relay. If you can use the fuel pump by-pass connector and start the truck or get fuel pump pressure what does that tell you?
Look at the drawing ....the relay is grounded in the same circuit up stream from the Air Flow Meter....
Now it could be the switch in the air flow meter because until it starts the ecu dosent give current to the sta terminal of the CO to make the fuel pump work. It could be bad connection at he Air Flow Meter connector,or even misadjusted spring plate .
Check the FC and E1 Connector at the Air Flow Meter make sure it is good when you crank ,or push the door a bit,for that matter make sure its moving while cranking.
Now if you guys are loosing residual pressure on your fuel rail ,then the fuel pump check valve may be bad ,clamp the line at the fuel rail coming from the pump see if it holds pressure or the fuel pressure regulator as stated could be leaking. Even the cold start injector might be the culprit.Or the fuel rail dampner could be the problem, the plastic cap on the top or back of the dampner can be removed to reveal a screw with a spring that isnt adjustable,but it does vibrate loose often and even back out.
We found this to cause a number of wierd problems at toyota like......hard starting hot soak as it is called.
There also is a vsv vacuum switch which aids in hot soak conditions . It takes the vacuum away from the pressure regulator to allow more pressure which in turns gives it just enough volume to aid in starting especially when you have a carboned up intake ......or cylinder combustion chamber.
If you can get the Air flow meter to ground when cranking as stated ,and still the fuel pump doesnt work ,but you can get the by-pass connector to work while cranking ,then check the wiring to the relay and the AFM .
Toyota efi parts rarely fail ...adjustments made need to be done yes and extreme use is another story but everyday driving is going to keep them running long into the junk yard.
Start with the basics as always and then work into the electronics.
Good luck
Gary
