A topology optimization method for maximizing the thermal performance of a heat sink with an axially uniform cross-section cooled by forced convection under the constraint of fixed pumping power is proposed. To this end, as the objective function, the total thermal resistance is suggested; this is inversely proportional to the thermal performance of the heat sink and consists of the convective resistance and the capacity resistance. An alternative mathematical expression for the total thermal resistance is proposed by deriving the relationship between the convective resistance and the thermal compliance. Compared to the conventional expression for the total thermal resistance, the proposed expression has an advantage in computation time required for sensitivity analysis of the total thermal resistance. The topology optimization method proposed in the present study is validated by comparing the geometry of a heat sink optimized by the proposed method with that of a plate-fin heat sink optimized analytically in a rectangular domain. Using the validated topology optimization method, a heat sink is optimized in a physical domain and simplified for manufacturability. The topology-optimized heat sink has a 31% lower total thermal resistance, as well as 9% reduced weight compared to the commercial heat sink. Hence, it is shown that the proposed topology optimization method can be used to design lighter heat sinks with higher thermal performance for practical applications. Finally, it is confirmed that minimizing the sum of the convective resistance and the capacity resistance is essential, and the omission of the latter may result in a heat sink design with poor thermal performance.