= Solution
An isotropic source's <bolometric luminosity> is distributed over a wavefront of physical area $4\pi a_0^2r^2$. Each <photon> loses an <energy> factor $(1+z)^{-1}$ through <cosmological redshift>, and successive <photons> arrive with a further rate factor $(1+z)^{-1}$ through <cosmological time dilation>. Therefore the <radiative flux> and <luminosity distance> are
$$
F=\frac{L}{4\pi a_0^2r^2(1+z)^2},
\qquad d_L=a_0r(1+z),\qquad 1+z=\frac{a_0}{a(t)}.
$$
These are bolometric formulas; a fixed observed <frequency> would require the shifted emission spectrum as well.
Normalize the <scale factor> by $a_0=1$. Summing the <radiative flux> over the <source counts on an FLRW past light cone> makes the transverse shell-area factors cancel:
$$
d\mathcal F_0=F\,dN=L n(t)a(t)^4dt,\qquad
\boxed{\mathcal F_0=L\int_0^{t_0}n(t)a(t)^4dt.}
$$
Thus $\mathcal F_0$ is power per effective detector area, summed over the whole sky; an all-sky detector of effective area $A_{\rm det}$ receives $A_{\rm det}\mathcal F_0$. The corresponding intensity per <solid angle> is $\mathcal F_0/(4\pi)$. Restoring the speed of light multiplies these flux integrals by $c$.
Spatially uniform $n(t)$ by itself does not specify its time evolution. For a persistent population comoving with the expansion, with neither creation nor destruction, the <cosmological continuity equation> for source number is $\dot n+3Hn=0$. Its conserved <comoving number density> is $n_0=a^3n$, so the <cosmological bolometric background from conserved sources> becomes $Ln_0\int a\,dt$. For the <Einstein-de Sitter universe>,
$$
\boxed{\mathcal F_{\rm flat}
=Ln_0\int_0^{t_0}\left(\frac{t}{t_0}\right)^{2/3}dt
=\frac35Ln_0t_0=\frac{2Ln_0}{5H_0}.}
$$
The last equality uses $H_0=2/(3t_0)$. The early-time <improper integral> converges despite the diverging physical source density. If instead the proper <number density> were maintained at a constant value $n_0$ by continuous source creation, the same geometry would give $3Ln_0t_0/11$. For a completely unspecified source history the general integral above is the determined answer; constancy of individual <luminosity> alone does not imply number conservation.
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