A mechanism of electron-mediated pumping of heat in the absence of net charge transfer is proposed. It may be realized in charge-neutral electron systems, such as graphene, coupled to an external electric potential. The flow of heat in this pumping cycle is not accompanied by a buildup of voltage along the system, which offers advantages over traditional thermoelectric cooling setups. The efficiency of heat pumping and the magnitude of heat flux are studied in the hydrodynamic regime for weak disorder. It is shown that the cycle efficiency may approach the Carnot limit. In a pristine system, even for an infinitesimal pumping potential, the heat flux remains finite. In particular, for a potential in the form of a traveling wave moving with velocity c, the pumping is perfect; the entire heat content of the electron liquid is advected with velocity c. For a general pumping cycle the heat flux is determined by the cycle geometry and disorder strength.