  Joint energy distribution of the bottom quark and antiquark from decays of
the top quark and antiquark produced in the reaction $e^+ e^- \to t \bar{t}$ is
studied. Main emphasis is put on $CP$-violation effects in the interaction of
the photon and $Z$ boson with the top quarks. Energy asymmetries of $b$ and
$\bar{b}$ quarks, which give access to the $CP$-violating terms, are
considered. To estimate the magnitude of these asymmetries, the $CP$-violating
$\gamma t t $ and $Z t t$ couplings are calculated in one-loop model with
exchange of the Higgs boson with the mass of $125$ GeV. Interaction of this
boson with the top quarks is assumed to include scalar and pseudoscalar
couplings. Values of these couplings are constrained from the recent CMS
analysis. Energy dependence of the asymmetries of $b$ and $\bar{b}$ quarks is
calculated up to $\sqrt{s}=1.2$ TeV and some interesting features of their
behavior are observed. These observables can be of interest for future studies
at electron$-$positron colliders CLIC and ILC.
