This study proposed a new design for the cold part of heat pipes that utilizes a Functionally Graded Material (FGM) in the heat sink with power-law fins. The heat transfer in the heat pipes was evaluated using the heat resistance method, and it was shown that the new design improves the performance of the heat pipe. A sensitivity analysis of geometrical and operational parameters was also conducted to examine their effects on the thermal management system. The results indicated that decreasing the inner radius and increasing the outer radius of the FGM fin with a power-law profile improves the thermal behavior by decreasing the absolute temperature of the fin by up to 1 Kelvin degree. It was also found that increasing the ambient air and the fin’s inner temperature had a negative effect on the fin’s performance by decreasing the amount of convection heat transfer and cooling. The study also showed that increasing the convection heat transfer coefficient and grading index reduces the absolute temperature by up to 1.5 Kelvin degrees, while decreasing the thickness profile coefficients, and conduction heat transfer coefficients positively affected the dimensionless temperature of the fin. The article concludes that using the new heat sink design improves the heat pipe’s performance and allows for the quick evaluation of the heat sink’s efficiency using an analytical solution.