Relationship between lifespan and reproductive age?
The sort of thing you're talking about would be called a Life History Invariant: a dimensionless ratio between two life history traits $A$ and $B$, such that although the traits themselves vary widely between species, the ratio $\frac{A}{B}$ is relatively constant.

Eric Charnov has written a lot about this, including a book.



One of the invariants which has been suggested is the ratio of age of maturity $\alpha$ to adult lifespan $E$. Charnov claims that it is relatively invariant within taxa, but has a different value for eg. fish vs. mammals vs. birds vs. reptiles.



From Charnov and Berrigan, "Dimensionless Numbers and Life History Evolution: Age of Maturity Versus the Adult Lifespan", Evolutionary Ecology (1990).

(If the ratio $\frac{A}{B}$ is invariant, then plotting $A$ vs. $B$ for different species on a regular graph will give you a straight line passing through the origin. The slope of the line gives you the value of the invariant.

Sometimes, in looking for an invariant researchers will instead graph the quantities on a log-log plot: they then look for whether the curve is a straight line with slope 1 (whose height gives the value of the invariant), with a good fit. This method has been criticized: the fact that some quantities are just about necessarily a fraction of other quantities (eg. you can't really lay an egg bigger than yourself), together with the fact that log-log plots "squish down" variance about the regression, opens the possibility of observing artifactual yesses. I tried to give an intuitive explanation of this criticism in a quora answer. Charnov and company have dismissed it as not applying to the invariants they found, however.)

This invariant is slightly different than the one you describe, in that they deliberately ignore pre-maturity deaths. (If infant mortality were high enough then it would be possible for average lifespan to be less than the age of maturity.) Also, they measure the ratio of pre-maturity lifespan to post-maturity lifespan $(\frac{\alpha}{E})$ instead of the the ratio of pre-maturity lifespan to total lifespan $(\frac{\alpha}{\alpha+E})$. These are just two different ways of talking about the same thing, but the former has an upper bound of 1, while the latter has no upper bound.

Finally, it's important to note that even if a quantity is relatively invariant, it can't be expected to be perfect; you have to decide how much noise you want to allow while still finding an approximate invariant interesting.