We perform large-scale numerical simulations of a directed version of the two-state stochastic sandpile model. Numerical results show that this stochastic model defines a new universality class with respect to the Abelian directed sandpile. The physical origin of the different critical behaviour has to be ascribed to the presence of multiple topplings in the stochastic model. These results provide new insight into the long-debated question of universality in Abelian and stochastic sandpiles.