Managing heat is a major challenge to meet future demands for a sustainable use of our energy resources. This requires materials, which can be custom-designed to exhibit specific temperature-dependent thermal transport properties to become integrated into thermal switches, transistors, or diodes. Common crystalline and amorphous materials are not suitable, owing to their gradual changes of the temperature-dependent thermal conductivity. We show how a second-order phase transition fully controls the temperature-dependent thermal transport properties of polymer materials. We demonstrate four major concepts based on a colloidal superstructure: (i) control of transition temperature, (ii) width of phase transition regime, (iii) multistep transitions, and (iv) step height of the transition. Most importantly, this unique control over thermal conductivity is only governed by the interparticle constriction, the particle composition, and its mesostructure. Our concept is therefore also applicable to a wide variety of other particulate materials.