Using the nonequilibrium Green's function method, we investigate the effects of edge reconstructions on the thermal conductance of graphene nanoribbons (GNRs). For the first time, we show that the thermal conductance of edge reconstructed zigzag GNRs oscillates with the number of dimers along the width direction, N. When N is an odd or even number, the thermal conductance of edge reconstructed zigzag GNRs is about 25 or 85% that of the perfect zigzag GNRs, respectively. For edge reconstructed armchair GNRs, the thermal conductance only decreases slightly and it changes linearly with N. The significant difference in the width dependence of GNR thermal conductance may be attributed to the topological symmetry and edge reconstruction of GNRs. Our results shed light on a new approach to control the thermal conductance of GNRs by reconstructing their edges and controlling their topological symmetry.