We report on the first thermodynamical investigation of (TMTTF)2Br in its antiferromagnetic ground state. There is a discontinuity in the lattice specific heat, Cp/T3, at 1.9-2.0 K which seems to reflect the existence of the Spin Density Wave (SDW) sub-phase. This discontinuity is similar to those we have measured in other quasi one-dimensional organic salts of the Bechgaard family. Above 2.5 K there is another anomaly which we attribute either to a pure vibrational lattice origin or/and to an additional configurational entropy, a manifestation of the disordered SDW ground state coupled to the lattice. As for a broad variety of density wave compounds, the specific heat of (TMTTF)2Br exhibits a contribution of low-energy excitations. They are at the origin of the time-dependent specific heat and complex energy-relaxation dynamics below 1 K. However, the corresponding time scale is much shorter than in the case of (TMTSF)2PF6, the Incommensurate SDW compound. We tentatively ascribe the difference in the kinetics of the energy relaxation of (TMTTF)2Br and (TMTSF)2PF6 to the different degrees of commensurability of their ground states.