If the volume produced by your headphones from the two systems isn't the same, the amplifier must be delivering different amounts of power. Increasing power delivery increases the heat produced. As an example, if one sounds sound twice as loud, the amplifier delivers ten times as much power. This is an exaggerated example because you'd notice the very obvious difference.
The volume and other settings may be the same on two systems but that's not the whole story. If you like, set the controls on your two computers so your headphones' loudness, measured by an SPL (Sound Pressure Level) meter, is the same from both. The amp's temperatures should then be the same.
You might like Wikipedia's articles on
Sound Pressure and
Decibel or you might not.