The stability of the BaCuOx (x≥2) phase has been mapped over a wide range of temperature (300–1100 °C) and oxygen pressure (10−5–103 bar). At ambient pressure and temperature, BaCuOx is found to be in a metastable state: long annealing at 450 °C tends to decompose the phase into Ba2Cu3O5 and BaO2. Having obtained the phase stability domain in the [T,p(O)2] plane we were able to prepare single-phase samples of BaCuOx with different oxygen contents suitable for precise intrinsic thermodynamical measurements. We show that the behavior of the low-temperature specific heat (1.1≤T≤32 K) and its dependence on the magnetic field (0≤B≤14 T) may be understood by taking into account a many-level magnetic system directly related to the Cu6O12 and Cu18O24 structural blocks of BaCuOx. Depending on the oxygen concentration, competition between antiferromagnetic (AF) ordering and the many-level system is observed. With increasing oxygen content, the Néel temperature decreases whereas amplitude of the many-level system increases. The zero-field AF transition belongs to the three-dimensional isotropic Heisenberg universality class.