= Solution
Take positive $\dot Q$ to mean net cooling per unit <mass>. At fixed <pressure> for an <ideal gas> of unchanged composition, $\rho\propto T^{-1}$, so $\dot Q=C T^{\alpha-1}-H$ with $C>0$. At <thermal equilibrium>, $CT_0^{\alpha-1}=H$. Because expansion work is included at fixed <pressure>, the perturbed <temperature> obeys
$$
c_p\frac{d\delta T}{dt}=-\left(\frac{\partial\dot Q}{\partial T}\right)_p\delta T
=-\frac{(\alpha-1)H}{T_0}\delta T.
$$
This <isobaric stability criterion for density-dependent power-law cooling> gives stability for $\alpha>1$ and instability for $\alpha<1$. For optically thin <bremsstrahlung>, $\alpha=1/2$, so
$$
\boxed{\frac{d\delta T}{dt}=\frac{H}{2c_pT_0}\delta T,\qquad\text{isobarically unstable}.}
$$
A slightly cooler parcel becomes denser and cools faster; a slightly hotter parcel has reduced cooling and heats further. The conclusion assumes the <sound-crossing time> is short enough to maintain nearly common <pressure>, with conduction and other stabilizing processes omitted.
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