I chased a very similar problem on an old leaf blower once. And it turned out to be the main fuel line from the tank to the carb. When it was cold, the line was fine. But after the engine ran for a bit and everything got warm, the old rubber line would get a little too soft and could collapse or get a tiny air leak when I tried to start it. It wasn't getting enough fuel. A new fuel line for a couple of bucks fixed it completely.
It might be something to check next time it acts up. When it's warm and won't start, just take a quick look at that fuel line to see if it looks soft or pinched.