Nothing to worry about. Fairly common to see at least a little browning of the sleeve there. If your issue is tied to heat, I would be looking at the clutch side oil seal and bearing. Also, what kind of shape is the chrome cylinder liner in? I have seen 044/046 saws that have the chrome nearly worn through from tons and tons of use. They'll start right up cold, but lose compression fast as they warm up. Some food for thought here...
Ignition problems: For the most part, it works or it doesn't (unless you have an old saw with points, then all bets are off. Not the case here). If an ignition problem is heat related, most of the time it will get hot and stop working. I have seen spark plugs get cracked and work ok when cold, but "short" when hot.
Air leaks caused by rubber parts: Will be evident immediately. Hot or cold, but will get worse as heat is applied and rubber softens.
Heat related issues: If it takes 10 minutes or so for the problem to manifest itself, likely there is a metal expansion problem. Look at piston/cylinder, or bearings. A bad clutch side bearing can make a saw do some really strange things. The heat from the failing bearing burns or softens the rubber in the crank seal and causes an air leak. Not an easy one to detect because of the oil pump and drive gear kind of help seal things up a bit. Especially when chock full of bar oil and sawdust. Once that cold thick bar oil that is always around that worm gear heats up and thins out, it's easier to pull air through the tiny gap between the worm and the pump housing.
If you can, acquire some block off plates and pressure/vacuum test that crankcase. If the plates aren't available, purchase or make some rubber pieces to install under the carb and muffler, and use a leakdown tester to apply slight air pressure to the crankcase. You'll probably hear the leak.
I hope I'm helping to steer you in the right direction.
