Why does your Briggs and stratton engine stop after it heats up?

Three things come to mind.

1) There is debris under the engine cover that is blocking the air flow which cools the engine. Once the engine gets overheated parts of the engine expand and do not operate properly. If it is hard to pull the rope to start the engine then it is probably this one.

2) The coil is going bad and breaking down when it gets hot. If the rope pulls easily then it is probably this one.

3) There might be an exhaust leak that is blowing onto the carburetor making the gas vapor lock.

It probably is not number 3. That one is pretty rare. 1 and 2 are very popular.