We develop a pathwise theory for scalar conservation laws with quasilinear
multiplicative rough path dependence, a special case being stochastic
conservation laws with quasilinear stochastic dependence. We introduce the
notion of pathwise stochastic entropy solutions, which is closed with the local
uniform limits of paths, and prove that it is well posed, i.e., we establish
existence, uniqueness and continuous dependence, in the form of pathwise
L1-contraction, as well as some explicit estimates. Our approach is
motivated by the theory of stochastic viscosity solutions, which was introduced
and developed by two of the authors, to study fully nonlinear first- and
second-order stochastic pde with multiplicative noise. This theory relies on
special test functions constructed by inverting locally the flow of the
stochastic characteristics. For conservation laws this is best implemented at
the level of the kinetic formulation which we follow here
