Why can we trust Hubble Time if the rate of expansion is not constant? The age of the Universe can be estimated from taking the inverse of the Hubble constant: $t_\text{universe} = 1/H_0 =d/v.$



It seems to me this method assumes that any given galaxy has been receding at a constant apparent velocity for the lifetime of the Universe. However, by including data from larger &amp; larger distances, it can be seen that Hubble's law doesn't hold. The data suggest that the rate of expansion was actually slower in the past.



Why is it appropriate to use only nearby galaxies to estimate the age of the Universe using the Hubble time when the rate of expansion isn't constant?
$H_0^{-1}$ is only a rough estimate for the age of the universe and you have correctly identified the reasons why not.

A correct age estimation relies on knowing $H_0$ and the densities of matter and dark energy, so that the past expansion history of the universe can be correctly modelled. Even this relies on an assumption about how dark energy behaves.

A more interesting question is why a value of $H_0 \simeq 70$ km s$^{-1}$/ Mpc gives an $H_{0}^{-1}$ of 14 billion years, which is within a few per cent of the current best estimate for the age of the universe of 13.8 billion years. In the $\Lambda$CDM cosmology model, the reason for this cosmic coincidence is that the universal expansion underwent a period of decelaraton up until about 4 billion years ago, when it started to accelerate again (the red curve on the plot below).

As a result, a tangent to the red curve, showing the size of the universe versus time, at the present epoch almost goes to ${\rm size}=0$ about 14 billion years ago. If we were to go backwards or forwards in time by 5 billion years, the agreement between $H_0^{-1}$ and the age of the universe would not be so good. At earlier times $H_0^{-1}$ would have overestimated the age of the universe, whereas at later times $H_0^{-1}$ will underestimate the age of the universe.