  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 microphysics) 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 microphysics 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.
