The specific heat of the quasi-one-dimensional organic compound (TMTSF)₂ClO₄ was measured between 1.9 and 35 K for two states corresponding to the metallic/superconducting ground state and the insulating spin-density-wave (SDW) ground state at low temperature, by varying the cooling conditions through the anion-ordering (AO) transition at 24 K. These are the first thermodynamical measurements of the SDW ground state. Like in the previously investigated compounds (TMTSF)₂X, with X = PF₆ and AsF₆, and (TMTTF)₂Br (where TMTTF≡tetramethyltetrathiafulvalene), the lattice specific heat shows strong deviations from a Debye behaviour, which can be explained by the quasi-one-dimensional structural character. Bumps in C/T³ occur at T = 3.5-4 K and 5 K for the quenched SDW and relaxed metallic states respectively; these are in good agreement with the frequencies of low-lying modes detected by infrared spectroscopy for the metallic sample. In addition, analysis of the lattice contribution suggests a T^α (α = 2.7) acoustic contribution for T≳2 K, ascribed to the low dimensionality of the structure. The SDW transition was detected at T_c = 4.5 K, as expected from the high value of the quenching rate (of 3.3 K s^-1) through the AO transition (the amplitude of the anomaly represents about 3% of the total heat capacity). The AO transition was investigated either by decreasing the temperature for the slow-cooled sample or by reheating, at a similar rate, for the quenched sample; the specific heat anomalies are identical for the two procedures, due to reordering annealing effects on reheating, a phenomenon previously investigated by x-ray experiments. Surprisingly, the entropy under the AO anomaly is Rln (4/3), much lower than the Rln 2 value expected for the two possible configurational orientations of the perchlorate ion in the disordered state. Similar low values of the entropy have been previously measured for the perrhenate (ReO₄) and BF₄ salts.