Amputees who use prosthetic limbs suffer from the problem of high contact temperature between the socket of the prosthetic limb and the amputated part and lack of evaporation of sweat. These conditions lead to discomfort and failure to perform functions properly. In addition, these conditions help generate ulcers and accumulate harmful bacteria in this area. This paper presents a heatsink design to extract heat from the contact area. A cylindrical heat sink is designed for phase-changing materials with three branched tubes in two stages. The current heat sink is used to cool the contact area between the amputated part and the socket in the lower prostheses. Three distributions of pipe branches are proposed. The distribution and pipe lengths were obtained using a constructal design method. In the constructal design, the lengths of the branched tubes were the degrees of freedom, the objective function was the minimization of the inlet temperature to the heat sink, and the constraint was the volume of the cylindrical heat sink. The metabolic heat transfer during exercise was estimated and its value was used to calculate the size of the cylindrical heatsink and the selection of the phase change material by testing three of them: water, tridecane, and dodecane. It was found that water gives the highest latent heat of melting and the lowest volume in addition to its availability. On the other hand, two cooling fluids were tested: water and air. It was found that water as a cooling fluid gave the lowest flow and the largest heat capacity. Constructal theory was used to design a cylindrical heat sink using branched tubes for the coolant in two steps: the first with three branches, and the second with nine branches. The degree of freedom for constructal theory was the length of the branches through the choice of their end locations. It was found that the branches of the highest length led to a reduction in temperature from 40°C to 15.48°C compared with the single tube, which reduced the temperature to 23.87°C. All tests recorded a pressure drop within the acceptable range of 3.1–5.43 Pa for the branches examined. The research demonstrated that using constructal theory achieved the best thermal dissipation within a restricted volume.