  During the standard ac lock-in measurement of the resistance of a
two-dimensional electron gas (2DEG) applying an ac current $I = \sqrt{2} I_0
\sin(\omega t)$, the electron temperature $T_e$ oscillates with the angular
frequency $2 \omega$ due to the Joule heating $\propto I^2$. We have shown that
the highest ($T_\mathrm{H}$) and the lowest ($T_\mathrm{L}$) temperatures
during a cycle of the oscillations can be deduced, at cryogenic temperatures,
exploiting the third-harmonic (3$\omega$) component of the voltage drop
generated by the ac current $I$ and employing the amplitude of the Shubnikov-de
Haas oscillations as the measure of $T_e$. The temperatures $T_\mathrm{H}$ and
$T_\mathrm{L}$ thus obtained allow us to roughly evaluate the thermal
conductivity $\kappa_{xx}$ of the 2DEG via the modified 3$\omega$ method, in
which the method originally devised for bulk materials is modified to be
applicable to a 2DEG embedded in a semiconductor wafer. The $\kappa_{xx}$ thus
deduced is found to be consistent with the Wiedemann-Franz law. The method
provides a convenient way to access $\kappa_{xx}$ using only a standard
Hall-bar device and the simple experimental setup for the resistance
measurement.
