= Solution
Let $\mathcal L$ be net cooling per unit volume, positive when energy is lost. At thermal equilibrium $\mathcal L=0$. For a slow perturbation that maintains <pressure> balance, the linearized thermal equation is
$$
\rho_0c_p\frac{d\delta T}{dt}=-\left(\frac{\partial\mathcal L}{\partial T}\right)_P\delta T.
$$
The heat capacity factor is positive. A positive derivative restores equilibrium: hotter gas cools more strongly, while cooler gas is heated. This is <isobaric thermal stability of a heated ideal gas>.
For fully ionized <hydrogen>, both protons and electrons contribute to the <ideal gas law>, so \b[$P=2nk_BT$]. Treat the stated constant heat input $\Gamma$ as a fixed heating rate per unit volume. Substituting $n=P/(2k_BT)$ into the cooling gives
$$
\mathcal L(T;P)=\frac{3P^2\alpha_B(T)}{8k_BT}-\Gamma.
$$
At equilibrium $T_0$, its derivative is
$$
\boxed{\left(\frac{\partial\mathcal L}{\partial T}\right)_{P,T_0}
=\frac\Gamma{T_0}\left(\frac{d\ln\alpha_B}{d\ln T}-1\right)_{T_0}.}
$$
With a temperature-independent recombination coefficient, the derivative is $-\Gamma/T_0$: \b[the equilibrium is thermally unstable to isobaric perturbations]. More generally, any nonincreasing $\alpha_B(T)$ also gives instability; stability requires its logarithmic slope to exceed one. Heating at fixed volume and pressure-balanced perturbations are distinct assumptions: the density falls as a parcel is heated, reducing the quadratic-density cooling.
No temperature law for $\alpha_B$ is supplied, so the displayed slope criterion states the general result without inventing one. A constant incident ultraviolet flux need not imply constant local volumetric heating in a self-consistent ionization-balance calculation. If the heating instead varies with density, temperature or neutral fraction, its derivative must be included; the conclusion here uses the explicit constant-$\Gamma$ model.
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