The efficient cooling of a finite-size volume generating heat, including adiabatic boundary conditions with the exception of a small heat sink, poses the problem of optimal allocation of high-conductivity material. Among the structural optimization methods, this article couples solid isotropic material with penalization parametrization (SIMP) with an aggregated objective function approach (AOF) to tackle this topology optimization problem through a multiobjective strategy. Both average and variance temperature-reduction problems is the solved by the identification of Pareto fronts, which are highly dependent on the quantity of the high-conductivity material. This study also underlines the link between the sensitivity analysis of both objective functions, which is required by the method of moving asymptotes (MMA). Furthermore, additional calculations have been done to include variations in heat-generation rate between two conductive materials by means of an additional penalization strategy. The main conclusion deals with the possibility of finding an acceptable trade-off between average and variance objective functions thanks to the convex shape of Pareto frontiers.