A cooled turbine runner for a gas turbine, in particular an aircraft engine, with turbine blades that are radially arranged at the circumferential surface of a rotor disk, wherein respectively a turbine blade is inserted with a profiled blade root into a correspondingly profiled disk finger groove at the circumferential surface of the rotor disk. A cooling device with at least one cooling air supply channel is provided, which extends at least substantially axially and at least over a part of the axial length of the blade root, and which has at least one inlet with an inlet opening at an inflow side of the blade root, wherein the inlet is embodied with a projection. A middle axis of the inlet has, in a direction oriented towards the inlet opening and in the area of the inlet opening, an extension component in the rotational direction of the rotor disk.