Overheating of a process liquid retained in a reservoir of a multiphase pump during extended gas slugs is avoided by circulating a cooling liquid in thermal contact with a process liquid through an external cooling apparatus, which can include a heat exchanger. In some embodiments, process liquid from the reservoir is circulated through the cooling loop, while in other embodiments a separate cooling liquid is circulated between a reservoir heat exchanger and the external cooling apparatus. The liquid in the cooling loop can be circulated by a separate cooling pump, or process liquid can be circulated through the cooling loop due to a pressure differential between an inlet and an outlet of the cooling loop within the multiphase pump. The multiphase pump can be a twin screw pump, and the reservoir can be formed between outer and inner casings of the multiphase pump.