  Suppose $(X,\omega)$ is a compact K\"ahler manifold. We introduce and explore
the metric geometry of the $L^{p,q}$-Calabi Finsler structure on the space of
K\"ahler metrics $\mathcal H$. After noticing that the $L^{p,q}$-Calabi and
$L^{p'}$-Mabuchi path length topologies on $\mathcal H$ do not typically
dominate each other, we focus on the finite entropy space $\mathcal E^{Ent}$,
contained in the intersection of the $L^p$-Calabi and $L^1$-Mabuchi completions
of $\mathcal H$ and find that after a natural strengthening, the $L^p$-Calabi
and $L^1$-Mabuchi topologies coincide on $\mathcal E^{Ent}$. As applications to
our results, we give new convergence results for the K\"ahler--Ricci flow and
the weak Calabi flow.
