The burner (1) for a gas turbine includes a duct (2) housing a plurality of tetrahedron shaped vortex generators (3) and a lance (4) to inject a fuel to be combusted. Within the duct (2), a plurality of vortex generators (3) are provided with a plurality of holes (9) for injecting cooling air. The cooling holes (9) define passing through areas that are non-uniformly distributed on a top wall (11) of the vortex generators (3). A method for locally cooling a hot gases flow passing through a burner includes non-uniformly injecting cooling air from a vortex generator into the hot gas flow in the duct, which can reduce the occurrence of flashback in the burner.