No, cooling is not the issue,. This is not a power transistor, its not meant to pump a lot of amps. Whats more once its turned ON its internal resistance is insignificant (225mΩ).

What can be happening here:

- badly designed output Audio section shorting power in some rare circumstances, might be as weird as mechanically stressed audio jack touching traces underneath it.

- software glitch around audio power enable routine enabling/driving that transistor hundred/thousand of times per second (pwm) in some circumstances, keeping it in the linear region

- badly designed under powered mosfet driver, either voltage too low or not enough current keeping it in the linear region

You do not cool power rail switches like this one, they arent meant to dissipate any meaningful power when designet properly.