Topology optimization uses a variable permeability approach to manipulate flow geometries. Such a method has been employed in the current work to modify the geometric configuration of internal cooling ducts by manipulating the distribution of material blockage. A modified version of the OpenFOAM solver AdjointShapeOptimizationFOAM has been used to optimize the flow-path of a serpentine channel and high aspect ratio rectangular ducts, with increase in heat transfer and reduction in pressure drop as the objective functions. These duct shapes are typically used as internal cooling channels in gas turbine blades for sustaining the blade material at high inlet temperatures. The serpentine channel shape is initially topologically optimized, the fluid path from which is post-processed and re-simulated in star-ccm+. The end result has an improvement in thermal performance efficiency (η) by 24%. Separation regions are found to be reduced when compared to the original baseline. The second test geometry is a high aspect ratio rectangular duct. Weight factors are assigned to the objective functions in this multi-objective approach, which are varied to obtain a unique shape for each such combination. The addition of mass penalization to the existing objective function results in a complex lattice-like structure, which is a different outcome in geometry and shape when compared to the case without any additional penalization. The thermal performance efficiency of this shape is found to be higher by at-least 18% when compared to the computational fluid dynamics results of a few other turbulator shapes from the literature.