By Start do you mean

A) it will not turn the engine over when you turn the ignition key
or
B) It turns the engine over but the engine won't fire?

From what you said where it would not turn engine over... it is the
starter... your brushes are worn inside that when enigne is hot, it will not
contact the starter... and when the starter cooled down... connection is
made...

So change the starter.

I'd lean heavily toward the brushes/windings in the starter. Try this:
next time the car won't start, turn on your head lights (or open the
door and look at your dome light) and watch what happens when you try
to start it. If lights fail to dim, that means no power is going to the
starter. That means an open circuit in the starter. New starter time.
If they dim to almost out (or are dim when you turn them on), chances
are you need to get the batt/charging system cked. A loose batt plate
will intermittently short and cause that problem. Having the starter
tested cold tells you very little. Never underestimate the power of a
loose battery post. Ck the easy stuff first.