The standard FRW metric with cosmic time is $$ ds^2 = -dt^2 + a^2(t)(\gamma_{ij}dx^i dx^j),$$ and we can measure $t$ as the proper time for comoving observers. Thus it makes sense to talk about the age of the universe $t_0$ in terms of the coordinate $t$.
When using conformal time, the metric becomes $$ds^2 = a^2(\eta)(-d\eta^2+\gamma_{ij}dx^idx^j).$$
Suppose I know the evolution of the scale factor $a(t)$ for all time, in fact lets just assume it's matter dominated, $a(t)\propto t^{2/3}$. Is there any way to make sense of the quantity $\eta_0$, the conformal time today? Is this quantity measurable? If not, and I compute an "observable" where $\eta_0$ appears explicitly, are there any ideas about what could have gone wrong?