Problem: Starter will not turn engine over when it is hot and sits about 30 minutes.

Lawn Machine 42" model 13bk608go62,
17hp Briggs & Stratton twin cyl engine #42a707-2238 made by MTD.

Starts and runs great on cold start. After using it about 45 minutes and I turn the engine off and let it sit about 30 minutes, the starter will not turn the engine over until it cools off for about 2 hours. The starter will try but just can't turn it over. It is the same sound you get when you have a weak battery. The battery is fully charged. All cables are clean and in very good condition. The engine does not appear to be too hot. I have replaced the solenoid and installed in-line fuel cut off valve. I have cut off the fuel valve and let the gas run out of fuel and it still will not turnover over when hot. The mower is always kept inside and clean. I only cut about ¾ acre. The mower is a well maintained and never been overused.

Time for a new starter...

You need to set your valve clearance.

Installed new starter with housing about a 1/4" larger and fix the problem. Now turn over like a horse rolling over in the dirt.

Thanks for the Follow-up. I actually experienced that first hand with an engine in that same family on my own Tractor 2 years ago, and several cars over the years.