High-porosity “cellular materials” are desirable for manufacturing multifunctional heat exchangers. While stochastic metal foams and regular strut-based topologies have been the focus of many prior investigations, there is very limited research on mathematically defined, triply-periodic minimal surfaces (TPMS). To this end, convective heat transport in sheet-based TPMS topologies, viz. gyroid and Schwarz P are numerically investigated and compared against “Tetrakaidecahedron,” which is a representative unit cell of stochastic metal foams. The candidacy of sheet-based topologies over strut-based in conjugate thermal transport applications was evaluated by plotting the product $\overline{h}{A}_{\mathrm{f}}$ against the pumping power, where gyroid outperformed the other two topologies.