•Optimal design of various periodic unit cells aiming at high thermal conductivity.•Schemes to modulate porous microstructure based on topology optimization.•Computational and experimental validation of effective thermal conductivity with error less than 10 %. Porous structures are lightweight and thus possess tailorable thermophysical properties through topological design. A multi-constraint topology optimization scheme is developed for designing 2D periodic lattice material with controllable porosity and optimal thermal conductivity. The porosity, pore size and specific surface area are modulated by imposing local density constraints, and thus the structure-determined thermal conductivity can be systematically investigated. The isotropic porous structures after optimization show effective thermal conductivity close to the Hashin-Shtrikman theoretical bound. The specific surface area of base cell is enlarged with imposing local density constraint. In addition, microstructure with anisotropic thermal conductivity can be also obtained. The samples of lattice structure are additively manufactured via selective laser melting, and the thermal conductivities are experimentally validated with deviation within 10 %. The proposed porous structures with targeted porosity have potential application in skeleton embedded with phase change materials.