= Solution
Take $z$ to increase inward, so the positive coefficient $A$ describes an inward-increasing <temperature>. Constant gravity and <hydrostatic equilibrium> give $dP/dz=\rho g$, hence
$$
\frac{dT}{dP}=\frac{A}{gT^3},\qquad \nabla_{\rm rad}=\frac{AP}{gT^4}.
$$
Matching the <radiative temperature gradient> to the <adiabatic temperature gradient> gives the formal local boundary relation
$$
\boxed{P_b=\frac{g}{A}\nabla_{\rm ad}T_b^4}.
$$
The same result follows from <radiative diffusion>: for constant upward thermal flux $F$ and <Rosseland mean opacity> $\kappa$, $A=3\kappa F/(16\sigma_{\rm SB})$.
There is an important limitation to treating $A$ as constant over the entire radiative layer. Integration from an irradiated outer boundary gives
$$
T^4=T_0^4+\frac{4A}{g}P,\qquad \nabla_{\rm rad}=\frac14\left(1-\frac{T_0^4}{T^4}\right)<\frac14.
$$
For a diatomic <ideal gas>, $\nabla_{\rm ad}=2/7>1/4$, so this profile cannot actually reach a <radiative-convective boundary>. Formally, imposing a constant $\nabla_{\rm ad}$ would give
$$
P_b=\frac{gT_0^4}{A}\frac{\nabla_{\rm ad}}{1-4\nabla_{\rm ad}},
$$
which is positive only for $0<\nabla_{\rm ad}<1/4$. \b[A finite boundary for a normal molecular atmosphere requires additional <opacity>, flux, or thermodynamic variation.] The local matching formula is usable near a real boundary, but constant $A$ is not a complete global model of it. This is the <convective stability of a constant-opacity irradiated atmosphere>.
For the intended order-of-magnitude scaling, suppose the local values of $g/A$ and $\nabla_{\rm ad}$ are comparable for <Jupiter> and a <hot Jupiter>, and assume $T_b$ scales with <planetary equilibrium temperature>. Equal absorbed-flux factors around the same stellar luminosity give $T_b\propto a^{-1/2}$. Taking $a_J=5.2\,\mathrm{AU}$ and a representative close-in orbit $a_h=0.05\,\mathrm{AU}$ yields
$$
\frac{P_{b,h}}{P_{b,J}}\sim\left(\frac{T_{b,h}}{T_{b,J}}\right)^4\sim\left(\frac{5.2}{0.05}\right)^2,
\qquad \boxed{P_{b,h}\sim10^3\,\mathrm{bar}}.
$$
This illustrates how irradiation can push a boundary much deeper. It is a conditional estimate calibrated from the supplied $0.1\,\mathrm{bar}$ reference, not a self-consistent prediction of the globally constant-$A$ model. Different intrinsic cooling flux, <opacity>, gravity, or <atmospheric metallicity of a giant planet> can substantially alter it.
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