Check the fuel filter and the sock filter in the tank. Could be there's a bunch of crud that stops up the sock and chokes off the fuel. I've seen them choke out and then when it sits a bit, the fuel backflows enough to unclog it and then the process starts over on the restart. It could be just coincidence that it takes the same amount of time to clog up as it takes to get to operating temp and the temp really has nothing to do with the problem.
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YO! Barry! You LOST!