An electric pump system and method of operating the same involves pumping a fluid through a fluid passageway defined in a mechanical pump from a pump inlet to a hollow shaft of a motor, through the hollow shaft to an internal motor cavity defined by a housing of the motor, and through another fluid passageway defined in the motor housing and mechanical pump that leads to a pump outlet. The system and method further involve pumping the fluid through another fluid passageway defined in the mechanical pump from yet another pump inlet to the pump outlet. The temperature of fluid exiting the hollow shaft can be assessed and used by an electronic control unit (ECU) of the electric pump system to control the same. The electric pump system can be part of a cooling and lubrication system for an electric vehicle transmission, gearbox, differential or transfer case, for example.