Prismatic cellular or honeycomb materials exhibit favorable properties for multifunctional applications such as ultralight load bearing combined with active cooling. Since these properties are strongly dependent on the underlying cellular structure, design methods are needed for tailoring cellular topologies with customized multifunctional properties. Topology optimization methods are available for synthesizing the form of a cellular structure—including the size, shape, and connectivity of cell walls and openings—rather than specifying these features a priori. To date, the application of these methods for cellular materials design has been limited primarily to elastic and thermoelastic properties, and limitations of classic topology optimization methods prevent a direct application to many other phenomena such as conjugate heat transfer with internal convection. In this paper, a practical, two-stage topology design approach is introduced for applications that require customized multifunctional properties. In the first stage, robust topology design methods are used to design flexible cellular topology with customized structural properties. Dimensional and topological flexibility is embodied in the form of robust ranges of cell wall dimensions and robust permutations of a nominal cellular topology. In the second design stage, the flexibility is used to improve the heat transfer characteristics of the design via addition/removal of cell walls and adjustment of cellular dimensions without degrading structural performance. The method is applied to design stiff, actively cooled prismatic cellular materials for the combustor liners of next-generation gas turbine engines.