Yep - typical symptoms of a failing battery. Its internal resistance rises, so it gets hot when you draw a lot of current from it. The voltage output drops and the phone either declares it as having a low charge, or just switches off immediately.
When you restart the phone it only draws a small current until you start gaming again. The small current from the battery means not much voltage is lost across its internal resistance, so the phone sees a healthy voltage and declares the battery to be well charged. Next time you start playing a game the current draw increases, the battery voltage drops, the battery gets hot, and it all repeats.
A new battery will fix it.
Note: the battery charge indication is only reliable when the battery is good and healthy. A failing battery will not give reliable charge readings.