  Multi-Higgs models equipped with global symmetries produce scalar dark matter
(DM) candidates stabilized by the unbroken symmetry. It is remarkable that a
conserved CP symmetry can also stabilize DM candidates, provided it is a CP
symmetry of order higher than two. CP4 3HDM, the three-Higgs-doublet model with
CP symmetry of order 4, is the simplest example of this kind. It contains two
mass-degenerate scalar DM candidates $\varphi$ and $\bar\varphi$, each of them
being a CP4 eigenstate and, therefore, its own antiparticle. A novel
phenomenological feature of this model is the presence of $\varphi\varphi
\leftrightarrow \bar\varphi\bar\varphi$ conversion process, which conserves CP.
It offers a rare example of DM models in which self-interaction in the dark
sector can significantly affect cosmological and astrophysical observables.
Here, we explore the thermal evolution of these DM species in the asymmetric
regime. We assume that a mechanism external to CP4 3HDM produces an initial
imbalance of the densities of $\varphi$ and $\bar\varphi$. As the Universe
cools down, we track the evolution of the asymmetry through different stages,
and determine how the final asymmetry depends on the interplay between the
conversion and annihilation $\varphi\bar\varphi \to $ SM and on the initial
conditions. We begin with the analytic treatment of Boltzmann equations,
present a detailed qualitative description of the process, and then corroborate
it with numerical results obtained using a dedicated computer code. Finally, we
check if the model can produce an observable indirect detection signal.
