= Solution
First specify the perturbation: take fixed <gas> <density>, fixed ionic abundance and the same <temperature>-independent heating, so thermal stability requires $d\mathcal C/dT>0$ at an equilibrium. Let $\theta=\Delta E/k_B$, $n_e$ be the collider <density> and $n_A=n_1+n_2$ the atomic <density>. A two-level optically thin balance is
$$
n_1n_eq_{12}=n_2(A_{21}+n_eq_{21}),\qquad\mathcal C=n_2A_{21}\Delta E.
$$
The <critical density of an atomic transition> is approximately $A_{21}/q_{21}$. At \b[low <density>], each rare collisional excitation is followed by an escaping <photon>, so
$$
\mathcal C\simeq n_An_eq_{12}\Delta E\propto T^{-1/2}e^{-\theta/T},\qquad\frac{d\ln\mathcal C}{d\ln T}=-\frac12+\frac\theta T.
$$
The <isochoric thermal stability of a collisionally excited line> therefore gives
$$
\boxed{T<2\theta:\ \text{stable};\qquad T>2\theta:\ \text{unstable}.}
$$
At $T=2\theta$ the linear restoring coefficient vanishes; the cooling maximum is marginal to first order and is not robust two-sided stability.
At \b[high <density>], collisional de-excitation competes successfully with radiative decay, and collisions establish <local thermodynamic equilibrium> level populations. Detailed balance gives $n_2/n_1=g e^{-\theta/T}$, where $g=g_2/g_1$. The cooling is then
$$
\mathcal C=n_AA_{21}\Delta E\,f_u(T),\qquad f_u=\frac{g e^{-\theta/T}}{1+g e^{-\theta/T}},\qquad\frac{d\ln f_u}{d\ln T}=\frac\theta T(1-f_u)>0.
$$
Thus \b[the high-<density> fixed-<density> branch is stable for finite $T$], becoming weakly restoring as the upper fraction saturates at high <temperature>. It is incorrect to retain the low-<density> factor $T^{-1/2}$ in the <radiative cooling> law once populations are thermalized.<AGN Emission Lines by Hagai Netzer>[Section2.2: collisional excitation, critical <density> and the high-<density> Boltzmann limit]{ref}
The <isobaric thermal stability of a two-level coolant> is a distinct question. If the intended heating is a fixed rate \b[per particle], then $n\propto T^{-1}$ and stability tests the cooling per particle. The low-<density> boundary becomes $T<2\theta/3$; the high-<density> cooling per particle remains proportional to $f_u$, so remains stable. If instead heating is constant \b[per volume], the isobaric low-<density> boundary is $T<2\theta/5$, and the high-<density> stability condition is $\theta(1-f_u)/T>1$. The source gives no <pressure> constraint or explicit heating normalization: the boxed $2\theta$ result is the fixed-<density> answer, with these conditional alternatives stated to avoid asserting a universal boundary.
\Image[/past-exam-of-the-formation-of-structure-in-the-universe-course-of-the-university-of-cambridge/2015/paper-1-line-cooling-stability.png]
{title=Low-density collisional line cooling has a temperature maximum, whereas high-density LTE emission rises toward saturation}
{source=paper-1-line-cooling-stability.py}
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