GaAs and GaN as low temperature carriers in transistor
Do you mean HEMT devices made using GaAs and/or GaN to be used in an amplifier circuit which is then operated at low temperatures?

What I can find regarding GaAs and GaN in combination with low temperatures mostly refers to the fabrication of GaAs / GaN devices where the growing of those layers is done at a (relatively) low temperature. This appears beneficial as the crystal structure becomes better (less defects) compared to growing them at high temperatures.

If you mean something else, please include a link to where you found that information so we can look into it.

Regarding the power dissipation. You appear to treat the power dissipation as a property of the transistor. Like: "This transistor always has that power dissipation." It does not work like that.

To build an amplifier, a (HEMT) transistor is used in a circuit where it is biased at a certain voltage and/or current". All this determines the power consumption that the transistor will have. It is for example possible to take a transistor rated for use at 100 W and use it at 1 W. The 100 W is the maximum power the transistor can cope with (provided it is adequately cooled) but it can also work at a much lower power.

So the circuit and way you use it determine the power consumption. That's why you cannot find a direct answer for the power dissipation. There is no direct answer as it depends on how you use it.