Cocrystals of cubane and fullerenes, C-60-cubane and C-70-cubane, show distinct rotational ordering transitions. We studied the corresponding structural changes with temperature-dependent X-ray diffraction and the thermodynamics of the phase transitions with adiabatic microcalorimetry and differential scanning calorimetry. C-60-cubane has one phase transition around 130 K from a high-temperature fcc phase with freely rotating C-60 to a low-temperature orthorombic phase in which the fullerene rotation is frozen. The corresponding enthalpy change is approximately 1170 J/mol, and the entropy change is 9.6 J/(mol K). C-70-cubane has two phase transitions. Around 380 K, the high-temperature fcc phase with freely rotating C-70 transforms into a bct phase in which the C-70 rotates uniaxially around an axis that precesses around the c direction with a full opening angle of 40 degrees. Around 170 K, the uniaxial rotation also freezes out, with an accompanying structural transition to monoclinic and enthalpy and entropy changes of 620 J/mol and 8.7 J/(mol K), respectively. The low-temperature specific heat was analyzed in terms of the Debye-Einstein model to estimate the librational energies of the fullerenes and Debye temperatures. We found very similar values for the two cocrystals, approximately E-lib = 2.2 meV and T-Debye = 23 K. For reference, we also measured the specific heats of pure C-60 and C-70 and found E-lib = 2.96 meV and TDebye = 32 K for C-60 and E-lib = 1.9 meV and T-Debye = 20 K for C-70.