If it was in my shop, I would check the compression. install a compression gauge and keep pulling the rope until the gauge stops going up. In most cases, if its around 120 when cold it will exhibit that "no start/hard to start when hot" characteristic, because the compression will be a bit lower when it is hot. One saw I recall was 120 cold and only dropped to 110 when hot, but that was enough to make a difference. It would not start until it cooled off.
If your compression is 130-155 or so the problem is probably something else.