The response of the heat release rate (flame transfer function, FTF) to a harmonic perturbation in air mass flow rate is studied with large eddy simulation in the case of partially premixed turbulent flames stabilized at the central recirculation zone (CRZ) generated via the mechanism of vortex breakdown in high swirl flows. The analysis is based on a previously developed methodology which, making use of the proper orthogonal decomposition (POD), can separate the FTF into three main components: the first driven by the equivalence ratio fluctuations, the second and the third by axial and tangential velocity fluctuations released at the swirler trailing edge. In the particular case of a free-standing CRZ, i.e., not anchored to a bluff body, it is found that the shape of the CRZ and the position of its apex are largely sensitive to small variations in dynamic head of the approach swirl flow at the centerline. Two main structures are found: type A) CRZ radially narrow with apex positioned inside the burner and type B) CRZ radially wide with apex positioned in the combustor. It is shown that the mechanism driving the flame response largely depends on the structure of the CRZ. More specifically, the amplitude of the FTF is found in the case of type B flow structure to be significantly larger than in the case of type A. On the other hand, FTFs with type A flow structure are characterized by phase distribution with higher sensitivity to variations in the equivalence ratio. This trend and behavior of the FTFs, which are also qualitatively identified in the experimental FTFs, are interpreted in terms of the coupling existing between flame and CRZ which is significantly different between the two vortex breakdown structures.