For $0\leq d\leq 1$, let $\lambda_d$ be the $d$-dimensional Hausdorff measure on $\mathbb{R}$. Note that it is translation-invariant. Does there exist a locally compact topology $\mathscr{T}_d$ on $\mathbb{R}$, finer than the usual topology and compatible with the (additive) group structure (i.e., $+$ and $-$ are continuous), such that $\lambda_d$ is, up to some normalization, the Haar measure for $(\mathbb{R},+,\mathscr{T}_d)$?

(For $d=1$ the usual topology provides a positive answer. For $d=0$ the discrete topology does. So the question is whether we can do something in between.)

Bonus points if $\mathscr{T}_d$ can somehow be made "canonical".