Metal organic frameworks (MOFs) and particularly copper benzene-1,3,5-tricarboxylate (HKUST-1) are excellent materials for gas storage (e.g., CH4, N2, H2 adsorption) and gas separation. In this work, reduced graphene oxide (RGO)–HKUST-1 mechanical mixtures were studied in order to reveal the effect of RGO content on the pressure tolerance of the texture and heat conductivity. HKUST-1 was obtained by two different synthesis routes. Air-dried MOF and RGO were thoroughly mixed prior to the compression. Powder XRD and Raman spectroscopy were used to characterize the response of the crystal structure, while low-temperature nitrogen adsorption was used the follow the adsorption properties of the pellets. Finally, the "flash" heat pulse method was used to assess the thermal properties. The gas adsorption isotherms revealed that the adsorption capacity decreases when RGO is added. Based on Raman and XRD results, we found that the synthesis route has an effect on multiple scales. We experimentally confirmed that evaluation of the thermal diffusivity requires a model more complex than the simple Fourier equation, due to the inherent heterogeneous structure of the material. A good approximation of the Fourier coefficient of thermal diffusivity was obtained using the parameters of the Guyer–Krumhansl equation. The heat pulse experiments also revealed possible size-dependent behavior.