The cooling system has got a problem, I would check it for any leaks if your are into fixing stuff yourself. If you find a leak it should be an easy fix and you can get back to mowing your grass. If there is no visual damage then the problem could be worser which involves stripping the cooling system and maybe part the engine. Hope this helped

Look for the oil drain bolt and make sure it is tight and check the oil levels frequently

The last sentence solved the problem. Its getting a vacuum in the gas tank. Some models have a valve on top of the gas cap that screws up and down. That is the vent and has to be open. As you use gas the area is displaced with air where the gas was. Maybe the mower does not have the correct cap to let it vent.

This sounds like you are getting vapor lock. You will want to replace the fuel cap. This should resolve your issue.

I have an Ariens mower with a 19.5 Briggs and Stratton. It starts up fine, will cut grass for about 10 minutes (until it gets hot) then won't start up again until it has cooled off. Oil was recently changed and has proper amount, tank has fresh gas. What I have found is if I take the gas cap off right as it starts to die it continues to run. What can I do to fix this?