This work is the first step towards understanding thermionic transport properties of graphene/phosphorene/graphene van der Waals heterostructures in contact with gold electrodes by using density functional theory based first principles calculations combined with real space Greens function formalism. We show that for monolayer phosphorene in the heterostructure, quantum tunneling dominates the transport. By adding more phosphorene layers, one can switch from tunneling dominated transport to thermionic dominated transport, resulting in transporting more heat per charge carrier, thus, enhancing the cooling coefficient of performance. The thermionic coefficient of performance for the proposed device is 18.5 at 600 K corresponding to an equivalent ZT of 0.13, which is significant for nanoscale devices.