A vacuum pump, in particular to a turbomolecular pump, has a pump inlet, a pump outlet and a pump space for a gas to be pumped arranged between the pump inlet and the pump outlet, and at least one cooling gas inlet for a cooling gas for cooling the vacuum pump, the pump further having one or more hollow regions for the cooling gas connected in a gas conducting manner to the cooling gas inlet and arranged outside the pump space, wherein the or each hollow region is bounded by at least one component of the vacuum pump to be cooled.