The size of the struts in nanoporous cellular solids typically has a secondary influence on the stiffness of the solids, but it leads to significant stiffening when it is on the same order as the higher-order material parameter. We examined this size-dependence using the higher-order finite-element method (FEM) in this study. Mathematical analysis showed that the displacement field that satisfies the conventional Lame equation can serve as a displacement field template in higher-order FEM. Benchmarking studies showed that results from simulations of beam bending and rod torsion using this FEM approach were in good agreement with results from analytical solutions and experiments. Using this approach, we showed that the stiffness of cellular solids is strongly affected by the cellular arrangement and the density when the cell size is on the order of the higher-order material parameter and that the stiffening behavior in nanoporous polyimide can be explained using higher-order theory. The FEM results also showed that a porous solid with half the weight can be engineered to become as stiff as a fully dense solid if the porous microarchitecture is tailored to take advantage of higher-order stiffening.