  Planck data provide precise constraints on cosmological parameters when
assuming the base $\Lambda$CDM model, including a $0.17\%$ measurement of the
age of the Universe, $t_0=13.797 \pm 0.023\,{\rm Gyr}$. However, the
persistence of the "Hubble tension" calls the base $\Lambda$CDM model's
completeness into question and has spurred interest in models such as Early
Dark Energy (EDE) that modify the assumed expansion history of the Universe. We
investigate the effect of EDE on the redshift-time relation $z \leftrightarrow
t$ and find that it differs from the base $\Lambda$CDM model by at least
${\approx} 4\%$ at all $t$ and $z$. As long as EDE remains observationally
viable, any inferred $t \leftarrow z$ or $z \leftarrow t$ quoted to a higher
level of precision do not reflect the current status of our understanding of
cosmology. This uncertainty has important astrophysical implications: the
reionization epoch - $10>z>6$ - corresponds to disjoint lookback time periods
in the base $\Lambda$CDM and EDE models, and the EDE value of $t_0=13.25 \pm
0.17~{\rm Gyr}$ is in tension with published ages of some stars, star clusters,
and ultra-faint dwarf galaxies. However, most published stellar ages do not
include an uncertainty in accuracy (due to, e.g., uncertain distances and
stellar physics) that is estimated to be $\sim7-10\%$, potentially reconciling
stellar ages with $t_{0,\rm EDE}$. We discuss how the big data era for stars is
providing extremely precise ages ($<1\%$) and how improved distances and
treatment of stellar physics such as convection could result in ages accurate
to $4-5\%$, comparable to the current accuracy of $t \leftrightarrow z$. Such
precise and accurate stellar ages can provide detailed insight into the
high-redshift Universe independent of a cosmological model.
