Cooling structures require sufficient thermal conductivity. However, structure with thermal conducting could suffer high temperature, and thermal deformation could become serious. Thus, designing structures suppressing thermal deformation is an important task for designing cooling structures. Structural characteristics like stiffness and thermal conductivity are affected by structural shape. Thus, we intend to design structure with sufficient thermal conductivity, small thermal deformation, and light weight. Since these design factors have a trade off relationship, this research aims to develop design method achieving these characteristics with high levels. Structural optimization methods are able to be utilized designing structure satisfying contradicting design factors. Topology optimization (TO) is one of the most flexible optimization methodology. Thus, TO is selected as structural optimization in this research. Since we intend to design structures have sufficient thermal conductivity and minimum thermal deformation, the target of optimization is maximization of stiffness for thermal stress and thermal conductivity is introduced as a constraint. As criteria of stiffness and thermal conductivity, the structural compliance and thermal compliance are used. The structural optimization is implemented using solid isotropic material with penalization (SIMP) method of TO. Design variables are updated by sequential linear programming (SLP) in the early stage. In the latter stage, phase field method is applied to update design variables. To clarify the validity and the utility of the proposed methodology, some numerical examples are studied. Through these numerical examples, optimal shapes with high thermal conductivity and high stiffness for thermal deformation are clarified.