  Finite volume methods are proposed for computing approximate pathwise
entropy/kinetic solutions to conservation laws with a rough path dependent flux
function. For a convex flux, it is demonstrated that rough path oscillations
may lead to "cancellations" in the solution. Making use of this property, we
show that for $\alpha$-H{\"o}lder continuous rough paths the convergence rate
of the numerical methods can improve from $\mathcal{O}(\text{COST}^{-\gamma})$,
for some $\gamma \in \left[\alpha/(12-8\alpha), \alpha/(10-6\alpha)\right]$,
with $\alpha\in (0, 1)$, to $\mathcal{O}(\text{COST}^{-\min(1/4,\alpha/2)})$.
Numerical examples support the theoretical results.
