Conductive heat transfer is of importance in the cooling of electronic equipment. However, in order for conductive cooling to become effective, the use of high-conducting materials and the correct distribution thereof is essential, especially when the volume which needs to be cooled has a low thermal conductivity. An emerging method of designing internal solid-state conductive systems by means of topology optimization is considered in this paper. In this two-dimensional study, the optimum distribution of high conductive material within a square-shaped heat-generating medium is investigated by making use of the “method or moving asymptotes” (MMA) optimization algorithm coupled with a numerical model. The use of such a method is considered for a number of cost (driving) functions and different control methods to improve the definiteness of the boundaries between the heat-generating and high-conduction regions. It is found that the cost function used may have a significant influence on the optimized material distribution. Also of interest in this paper are the influences of thermal conductivity and the proportion of the volume occupied by the high-conducting solid on the resulting internal cooling structure distribution and its thermal conduction performance. For a square domain with a small exposed isothermal boundary centered on one edge, a primary V-shaped structure was found to be predominantly the most effective layout to reduce the peak operating temperature and to allow for an increase in the internal heat flux levels.