<<Sounds like George Braly has the answer.  It is the fuel pump!  Well, I
don't know about fuel pump or not, but I know that a hot engine just needs
more air to get started, fuel pump problem or not.  I have never had a
problem starting my engine when it is hot, and I don't know if there is a
problem with my fuel pump or not.  I have never "cooled my fuel pump."  You
just push the throttle to at least half or more open, mixture to idle cut
off and crank away.>>

It would be the first time I disagree with George, but I don't see how it
could be the fuel pump.  The only thing the fuel pump could do wrong is to
not pump.  I installed a fuel pressure gauge on my C177 between the pump and
the fuel servo.  I found that the fuel pump would hold pressure for some
time after shutdown - probably an hour or so.  Also, if the fuel pump
weren't working the electric pump would, especially the one in my Cessna as
it is under the floor and stays cool.  The primary symptom is that the
engine will start with the mixture off.  That means no fuel is allowing into
the engine so it doesn't matter whether or not the pump is working during
cranking.  This is all about the Precision Airmotive system on Lycomings,
not the Continental system.

Gary Casey