This paper studies a family of finite volume schemes for the hyperbolic scalar conservation law $u_t +bla_g \cdot f(x,u)=0$ on a closed Riemannian manifold M. For an initial value in BV( M) we will show that these schemes converge with a $h^{\frac{1}{4}} $ convergence rate towards the entropy solution. When M is 1-dimensional the schemes are TVD and we will show that this improves the convergence rate to $h^{\frac{1}{2}}.$.