= Solution
Use the dark-sector <temperature> in every dimensionless integration variable: $x=m_\phi/T_\phi$ and $y=p/T_\phi$. Assume a single real <scalar field>, $g=1$, and fast number-changing reactions keeping $\mu_\phi=0$. For $x\gg1$, the populated momenta satisfy $y=O(\sqrt x)$, so
$$
\sqrt{x^2+y^2}=x+\frac{y^2}{2x}+O\left(\frac{y^4}{x^3}\right),\qquad
\frac1{e^{\sqrt{x^2+y^2}}-1}\simeq e^{-x-y^2/(2x)}.
$$
This is the nonrelativistic <Maxwell-Boltzmann distribution> limit of the <Bose-Einstein distribution>. The given Gaussian <Gamma function> integral gives
$$
\int_0^\infty y^2e^{-y^2/(2x)}\,dy=\frac{\sqrt\pi}{4}(2x)^{3/2},\qquad
n\simeq\left(\frac{m_\phi T_\phi}{2\pi}\right)^{3/2}e^{-x}.
$$
Since $\rho_\phi=m_\phi n+3nT_\phi/2+\cdots$ and $P_\phi=nT_\phi$, the <entropy density at zero chemical potential> is $s_\phi=n(x+5/2+\cdots)$. Its leading <nonrelativistic entropy at zero chemical potential> is therefore
$$
\boxed{s_\phi\simeq\frac{m_\phi^3}{(2\pi)^{3/2}}x^{-1/2}e^{-x}.}
$$
An internal multiplicity $g$ would multiply this expression; the absence of $g$ in the printed result assumes a real one-state scalar.
Now use the constant <separately conserved visible and dark entropy> ratio and the relativistic visible-bath entropy law, $s_{\rm SM}=(2\pi^2/45)g_{*s}^{\rm SM}T_\gamma^3=\xi s_\phi$. Since $T_\phi=m_\phi/x$,
$$
\left(\frac{T_\gamma}{T_\phi}\right)^3\simeq\frac{45}{2\pi^2(2\pi)^{3/2}}\frac\xi{g_{*s}^{\rm SM}}x^{5/2}e^{-x}.
$$
Thus the <cannibal-sector temperature ratio> has
$$
\boxed{\frac{T_\gamma}{T_\phi}\simeq k\left(\frac\xi{g_{*s}^{\rm SM}}\right)^{1/3}x^{5/6}e^{-x/3},\qquad k=\left[\frac{45}{2\pi^2(2\pi)^{3/2}}\right]^{1/3}.}
$$
The numerical coefficient is approximately $0.525$. The visible-bath temperature is the photon temperature while that bath is internally thermalized; after distinct visible species acquire different temperatures, its total entropy must instead use the appropriate temperature-weighted $g_{*s}^{\rm SM}$.
The displayed relativistic scalar entropy formula must not be continued into this regime with $g_{*s}^\phi=1$ held constant. If it is used as an effective definition at arbitrary temperature, then
$$
g_{*s}^{\phi}(x)\simeq\frac{45}{2\pi^2(2\pi)^{3/2}}x^{5/2}e^{-x},
$$
and $T_\gamma/T_\phi=(\xi g_{*s}^{\phi}/g_{*s}^{\rm SM})^{1/3}$ gives the same result. This specifies how the relativistic expression can legitimately be combined with the nonrelativistic one.
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