Let $\mathbb{Q}_n$ denote the $n$-th layer of the cyclotomic $\mathbb{Z}_p$-extension over the rationals, i.e. the unique real subfield of the cyclotomic field $\mathbb{Q}(\zeta_{p^{n+1}})$ of degree $p^n$ over $\mathbb{Q}$ for odd primes $p$.
We know that the cyclotomic units $C_{p^n}$ of $\mathbb{Q}(\zeta_{p^n})$ are of finite index in the full unit group $E_{p^n}$. We know (from Theorem 8.2 and exercise 8.5 Washington's book on Cyclotomic Fields) that the index $[E_{p^n}: C_{p^n}]= [E_{p^n}^+ : C_{p^n}^+]= h_{p^n}^+$. Here, we use the standard notation $h_{p^n}^+$ for the class number of $\mathbb{Q}(\zeta_{p^n})^+$, the totally real subfield of $\mathbb{Q}(\zeta_{p^n})$.
If we use the notation, $C_n= C_{p^n}\cap \mathbb{Q}_n$ and $E_n= E_{p^n}\cap \mathbb{Q}_n$. What is the index $[E_n: C_n]$?
I eventually found that what I was looking for is the main theorem in a paper by Sinnott from 1980. https://link.springer.com/content/pdf/10.1007/BF01389158.pdf