Graded porous structures can improve the design performance with a smooth transition of material properties over the domain. We present a multi-physics topology optimization approach for designing functionally graded porous structures. As an application, we present a formulation to optimize heat dissipating structures to enhance structural and thermal performance. The pore size, resolution, and porosity of the structures can be controlled to obtain optimized mechanical and thermal performances. Several controlling schemes are introduced which can tailor the design of the porous structures. Design approaches based on the functional variation of porosity, pore size and sensitivity-based porosity control are employed. Numerical examples with different structural loads, thermal loads and boundary conditions are presented to verify the formulation and show the efficacy of the implementation. We also investigate the effect of the controlling parameters on the overall design. The proposed formulation shows good promise to design functionally graded porous structures with targeted porosity in multi-physics systems. Display omitted •A design approach to generate porous structures guided by multi-physics topology optimization is presented.•Schemes to design functionally graded controllable porous structures based on porosity and pore size gradation are introduced.•A sensitivity-based controlling scheme is adopted to obtain physics based graded porous structures.