I have found that in unstructured mesh, discretizing the laplacian operator with finite volumes requires special care, as given in An Introduction to Computational Fluid Dynamics: The Finite Volume Method, (section 11.8) but this is limited to constant normals along the element faces. What happens when these faces are curves as in high order geometries? I haven't been able to find methods to get around this.
I have work in the FVM for many years. The second order FVM is very popular in the CFD area. The integration is approximated with the mid-point integration method. It means that the curved edge should be approximated with many line segments. If you want to work with the high-order FVM method, I suggest you try the spectral-FVM method.
In finite volume methods, you just have to figure out to parameterize the fluxes across the faces that are now curved. In finite element methods, this is a standard approach and not complicated at all. Most finite element libraries allow you to choose higher order geometry descriptions.