Richard,
Have you checked to see if the fuel tank vent line is clear and not clogged with spider debris? Your symptoms sound consistent with a fuel pump that is able to draw fuel in the beginning, but the longer it runs with a clogged vent line the tank will form a vacuum, and the fuel pump will not be able to pump the fuel. Anyway, something that’s easy to check and cheaper than replacing the fuel pump. I don’t know about the temperature other than to ask are you sure the right thermostat is in the engine?
Jim Cosgrove
FATE 30U #343
Galesville MD