Topology optimization is vital in the conceptual design of various functional devices. While for thermal fluid problems, it is challenging to manage the trade-off between maximizing heat exchange and minimizing power consumption. This paper studies the individual, summation, and logarithmic summations of two conflicting objective functions in a self-developed topology optimization program for a liquid-cooled cold plate design problem. The optimized results show that the logarithmic summations provide good convergence toward the dual-objective numerical oscillations, generating smooth and continuous channel layouts. Meanwhile, it can be extended to customized layouts under the control of weighted factors. A comparison study is conducted with a conventional parallel-cooling cold plate design to validate the optimized design. Higher heat transfer coefficients and lower pressure drop are observed from the simulation results, thereby highlighting the superiority of the optimized design.