A rotor and stator cooling system for a rotary vane pumping machine having two end plates, a stator assembly, and a rotor. A rotor cooling gas supplied at a cooling gas supply channel in an end plate passes from a radial inner location, along a rotor face chamber of the rotor in an outward radial direction, and then toward a plurality of rotor gas channels in the rotor. The rotor cooling gas absorbs heat from the rotor and then exits through a heated gas exit channel in another endplate. A stator cooling fluid entering at a cooling fluid port in one end plate passes through stator fluid channels of the stator assembly, absorbs heat therein, and exits at another fluid port in the other endplate.