  Recent results for the Ising model with first ($J_1$) and second ($J_2$)
neighbour interactions on the body-centered cubic (bcc) lattice suggest that
this model can host signatures of strong frustration, including Schottky
anomalies and residual entropy, as well as, a spin-liquid-like phase [E.
Jur\v{c}i\v{s}inov\'a and M. Jur\v{c}i\v{s}in, Phys. Rev. B, 101 214443
(2020)]. Motivated by these findings, we investigate phase transitions and
thermodynamics of this model using a cluster mean-field approach. In this
lattice, tuning $g=J_2/J_1$ leads to a ground-state transition between
antiferromagnetic (AF) and superantiferromagnetic (SAF) phases at the
frustration maximum $g=2/3$. Although the ordering temperature is reduced as $g
\to 2/3$, our findings suggest the absence of any Schottky anomaly and residual
entropy, in good agreement with Monte Carlo simulations. We also find a direct
transition between AF and SAF phases, ruling out the presence of the
spin-liquid-like state. Furthermore, the cluster mean-field outcomes support a
scenario with only continuous phase transitions between the paramagnetic state
and the low-temperature long-range orders. Therefore, our results indicate the
absence of strong frustration effects in the thermodynamics and in the nature
of phase transitions, which can be ascribed to the higher dimensionality of the
bcc lattice.
