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Covalent organic frameworks (COFs) are an emerging class of polymeric crystals with immense permanent porosities and large surface areas that are highly sought after for a variety of applications such as in heterogeneous catalysis, flexible electronics, and gas separation and storage. Although efficient heat dissipation plays a critical role in such applications, thermal transport in COFs has frequently been overlooked or has been an afterthought. Herein, we investigate the heat transfer mechanisms in the prototypical 3D COF-300 and demonstrate that interpenetration of the organic frameworks can result in dramatic enhancements in the heat transfer efficacies facilitated through supramolecular interactions between the individual frameworks. More specifically, we show that a 3-fold interpenetration can lead to as much as 6-fold increase in the room temperature thermal conductivity of COF-300. We attribute this to phonon hardening and reduced vibrational scattering at the pores resulting from the reinforcement due to supramolecular interactions that rigidify the individual frameworks in the interpenetrated structures. We also show that these materials possess ultralow and modular elastic moduli, thus positioning them as materials that are both lightweight and mechanically flexible, all the while maintaining relatively high thermal conductivities. [ABSTRACT FROM AUTHOR]