Let $\mathfrak{g}$ be a semisimple real Lie algebra and $G$ its associated adjoint Lie group. If we set $\kappa$ to be its Killing form, and $\theta$ to be some Cartan involution, we get a scalar product $$\kappa_\theta(x,y) := -\kappa(x,\theta(y)) \quad \forall x,y \in \mathfrak{g}$$ and hence a notion of a unit sphere $B_1(0) \subset \mathfrak{g}$.
Further, if $\nu \in \mathfrak{g}^*$ is any element, write $O_\nu := \text{Ad}_G^* \nu$ for its coadjoint orbit. This has a canonical volume form via the KKS symplectic form, and since the Killing form of a semisimple Lie algebra $G$-equivariantly identifies $\mathfrak{g} \cong \mathfrak{g}^*$, the adjoint orbits $O_X$ for $X \in \mathfrak{g}$ also are equipped with a canonical volume form. If we fix a semisimple $X \in \mathfrak{g}$, then the orbit $O_X$ is closed in $\mathfrak{g}$ and hence $B_R(0) \cap O_X$ is compact, so it makes sense to talk about the volume of the set $B_R(0) \cap O_X$ as a subset of $O_X$.
Question: What do we know about the scaling behaviour of the volume of $B_R(0) \cap O_X$ for different values of $R$? In other words, if $$\text{Vol}(B_1(0) \cap O_X) \neq 0,$$ what do we know about the function $f : [0,\infty) \to [0,\infty)$ with $$\text{Vol}(B_R(0) \cap O_X) = f(R) \cdot \text{Vol}(B_1(0) \cap O_X)?$$
My own thoughts: For nilpotent orbits, this question has an elementary answer, since nilpotent orbits themselves are cones, i.e. $r \cdot O_N = O_N$ for positive $r$, so $$B_R(0) \cap O_N = R \cdot (B_1(0) \cap O_N),$$ so that the volume $\text{Vol}(B_R(0) \cap O_N)$ with respect to the KKS volume form only depends on the scaling behaviour of the KKS form. If I remember correctly, if $n$ is the dimension of the orbit, then for nilpotent orbits my question is answered affirmatively with $f(R) = R^{n/2}$. (Note that here, one needs some additional argument for why the volumes of the balls are finite, since nilpotent orbits aren't necessarily closed and hence their intersections with $B_R(0)$ are not necessarily compact, but there's some ways to get there, either by some singularity-resolution arguments or other tricks from the literature).
For other kinds of orbits, this will probably be more tricky. The function $f$ must be zero in a whole neighbourhood of $0$, since non-nilpotent orbits $O_X$ have a nonzero distance from $0 \in \mathfrak{g}$, hence $B_R(0)$ will not intersect the orbit for $R$ in a whole neighbourhood of $0$.