  We analyse the thermal evolution of the Universe in the Inert Doublet Model
for three regions of Dark Matter masses: low mass (4-8) GeV, medium mass
(30-80) GeV and high mass (500-1000) GeV. Those three regions of DM mass
exhibit different behaviour, both in the possible types of evolution and in the
energy relic density values. In this analysis we use the masses of the scalar
particles as the input parameters to constrain the two self-couplings between
neutral scalars: $\lambda_{345},\lambda_2$. These couplings are used to
construct the parameter space where different types of evolution may be
presented. We discuss the influence of the scalar masses on the type of the
evolution. We also discuss the importance of the $\lambda_2$ self-coupling. We
argue, that the astrophysical data along with the positivity constraints
simultaneously constrain both $\lambda_{345}$ and $\lambda_2$ self-couplings.
