  We present measurements of the Joule heating of a 2D hole gas (2DHG) formed
in a 30nm GaAs quantum well. The hole density is in the range
(4.6-18.9)*10^9cm^-2 and exhibits an apparent metal-to-insulator transition
(MIT) with a critical density 6*10^9 cm^-2. In the limit of zero heating power
density P, the GaAs lattice is within 2 mK of the 6 mK base temperature of our
dilution refrigerator determined by He-3 melting curve thermometry. Throughout
the range of heating power densities used (1 to 10^6 fW/cm^2), the temperature
rise of the lattice is estimated to be negligible compared to the temperature
rise of the hole gas. We argue that the hole scattering rate is only a function
of the hole temperature, with little dependence on the lattice or impurity
temperatures in the relevant temperature range below 150 mK. We have therefore
made measurements of the hole resistivity at negligible heating power density
(P<5fW/cm^2) as a function of measured lattice temperature in the range 6 to
150 mK. We then use the hole resistivity measured in a cold lattice to estimate
the temperature of the 2D hole gas as a function of P. In the low hole density
insulating phase, the heating power density P(T) that heats the hole gas to a
temperature T exhibits a dependence P ~ T^2 for T<30mK, gradually changing to P
~ T^4 for higher temperatures. On the metallic side of the MIT, P is
proportional to T^5 or T^6 with a magnitude roughly 4 times less than the power
density required to heat the insulating phase to the same temperature. Our
measurements are within a factor of two of the available quantitative
theoretical predictions for the hole energy loss rate as a function of
temperature.
