In an axial flow gas turbine (30), a substantial reduction of the consumption of cooling air can be achieved by providing, within a turbine stage (TS), structure (39-44) to reuse the cooling air that has already been used to cool, especially the airfoils of, the vanes (33) of the turbine stage (TS), for cooling the stator heat shields (38) of that turbine stage (TS) downstream of the vanes (33).