= Solution
Use the observed present <photon> <temperature> $T_{\gamma0}\simeq2.725\,\mathrm K$. This is an additional observational input, consistent with <The Temperature of the Cosmic Microwave Background by Dale J. Fixsen>[abstract: combined CMB-temperature determination]{ref}. Standard <neutrino decoupling> and electron-positron <entropy> transfer give
$$
\boxed{T_{\nu0}=\left(\frac4{11}\right)^{1/3}T_{\gamma0}\simeq1.945\,\mathrm K.}
$$
For nonrelativistic relic <neutrinos> this is the <temperature> parameter of the redshifted momentum distribution, not the <temperature> of a newly thermalized gas of massive particles.
The <photon number density> from part <4/b> is
$$
n_{\gamma0}=\frac{2\zeta(3)}{\pi^2}
\left(\frac{k_{\rm B}T_{\gamma0}}{\hbar c}\right)^3\simeq410.5\,\mathrm{cm^{-3}}.
$$
One <neutrino> mass eigenstate and its antineutrino have two populated <helicity> states. Thus
$$
\boxed{n_{\nu_i}+n_{\bar\nu_i}
=\frac34\left(\frac{T_\nu}{T_\gamma}\right)^3n_\gamma
=\frac3{11}n_\gamma\simeq112\,\mathrm{cm^{-3}}.}
$$
The total over three species is about $336\,\mathrm{cm^{-3}}$, including antineutrinos. Comoving number conservation preserves these abundances after the particles become nonrelativistic.
For the supplied $H_0=72\,\mathrm{km\,s^{-1}\,Mpc^{-1}}=2.33\times10^{-18}\,\mathrm{s^{-1}}$,
$$
\rho_c=\frac{3H_0^2}{8\pi G}\simeq9.74\times10^{-27}\,\mathrm{kg\,m^{-3}}.
$$
Assuming nonnegative other energy components, relic <neutrino> rest-mass density cannot exceed the near-critical total density. With the common per-species abundance $n_*=n_{\nu_i}+n_{\bar\nu_i}$,
$$
\rho_{\nu,\rm rest}=n_*\sum_{i=1}^3m_i\le\rho_c,
\qquad
\boxed{\sum_{i=1}^3m_i\lesssim\frac{\rho_c}{n_*}
\simeq48.8\,\mathrm{eV}/c^2\approx50\,\mathrm{eV}/c^2.}
$$
Using the total $336\,\mathrm{cm^{-3}}$ would instead bound the mean mass, not the sum. This <critical-density bound on the sum of relic neutrino masses> is deliberately conservative: baryons, dark matter and <dark energy> leave less density available. It assumes the standard thermal abundance, negligible <chemical potentials>, no later dilution or extra populated states, and the idealized decoupling count. Small non-instantaneous-decoupling corrections do not matter for this estimate.
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