If you turn on and off the game immediately after it as warmed up and the sound is still there, but then after turning it off and letting it cool off for a while the sound returns, that would likely be a temperature intermittent. Give the cables the first examination & also check all your power supply voltages.

If it's not the cables or power supply, I would then get the freeze mist out and spray components one at a time in the audio output chain to see if a specific component or area of the audio power amplifier is sensitive to the temperature and begins to work again.

As Xenon75 said, you can feel the ICs. On some electronics certain ones do get warm, not sure about these specifically. If you pull away a finger because it burns, yeah that's surely bad. But also it might not be an IC. Could be a different component (regulator, cap, transistor, resistor etc..) that could be failing when it gets warm. A failing component begins to pass more current, this makes it get hotter which in turn makes it pass more current, and so on. The formal name for this is "thermal runaway"

The freeze spray is a much more controlled and intense cold, so not certain that the compressed air will work but you can certainly try it if that is all you have access to. Another thing we do with a temp failure that takes some time before it happens is the reverse, hitting components with the heat gun a little to accelerate the temperature. This is a lot more difficult and a lot more risky though.

Good luck, let us know.