Thermodynamic investigation by calorimetric measurements of the layered organic superconductors, κ-(BEDT-TTF)2Cu[N(CN)2]Br and its partially deuterated compounds of κ-(d[2,2]-BEDT-TTF)2Cu[N(CN)2]Br and κ-(d[3,3]-BEDT-TTF)2Cu[N(CN)2]Br, performed in a wide temperature range is reported. The latter two compounds were located near the metal–insulator boundary in the dimer-Mott phase diagram. From the comparison of the temperature dependences of their heat capacities, we indicated that lattice heat capacities of the partially deuterated compounds were larger than that of the pristine compound below about 40 K. This feature probably related to the lattice softening was discussed also by the sound velocity measurement, in which the dip-like structures of the Δv/v were observed. We also discussed the variation of the electronic heat capacity under magnetic fields. From the heat capacity data at magnetic fields up to 6 T, we evaluated that the normal-state γ value of the partially deuterated compound, κ-(d[3,3]-BEDT-TTF)2Cu[N(CN)2]Br, was about 3.1 mJ K−2 mol−1. Under the magnetic fields higher than 3.0 T, we observed that the magnetic-field insulating state was induced due to the instability of the mid-gap electronic state peculiar for the two-dimensional dimer-Mott system. Even though the volume fraction was much reduced, the heat capacity of κ-(d[3,3]-BEDT-TTF)2Cu[N(CN)2]Br showed a small hump structure probably related to the strong coupling feature of the superconductivity near the boundary.