Are the unit partial quotients of $\pi, \log(2), \zeta(3) $ and other constants $all$ governed by $H=0.415\dots$?
A probability distribution of the continued fraction expansion terms follows the Gauss-Kuzmin distribution for almost all irrational numbers:
$$p(k)=-\log_2\left(1-\frac{1}{(k+1)^2}\right)=\log_2\frac{(k+1)^2}{k(k+2)}$$
All generalized Khinchin's constants (including $K=K_0$, the geometric mean), are derived from this distribution.
In this case, you seek the fraction of terms with value $1$, which is
$$p(1)=\log_2 \frac{4}{3}\approx 0.4150375$$
So it turns out this constant you observed is expressible with elementary functions.