why does the engine stall after startup and run again after a one or more hours

What is your question?

The engine runs for a few seconds and then stalls, it starts one more time and quits, after that it will not fire again. It sits then for an hour or more. It then starts again and operates without fail for several months, until one day, after making a stop at a place the motor starts- starts and then will not fire up again until it has a rest.

What makes this problem better or worse?

Weather conditions are irrelavant

How long has your 2009 Pontiac Vibe had this problem?

It happened three times over a one year period.