  In this work, we study a pseudo-Nambu-Goldstone boson (pNGB) dark matter
model extended with a real scalar. The dark sector is assumed to be feebly
coupled with the standard model (SM) via a Higgs portal, so that the pNGB dark
matter is produced by the freeze-in mechanism. Since the production happened in
a very high energy era, we introduce an extra scalar field which is weakly
coupled to the SM for stablizing the electroweak vacuum. Our model can
reproduce the correct relic abundance of dark matter favored by observations.
In addition, we determine the evolution of couplings in higher energy scale by
solving the renormalization group equations, and show that the self coupling of
the real scalar, $\lambda_S$, and the mixing coupling between the Higgs field
and the real scalar, $\lambda_{HS}$ are stringently constrained by the
conditions of vacuum stability and couplings perturbativity up to the Planck
scale. We also find that the relic abundance of DM is insensitive to the values
of $\lambda_S$ and $\lambda_{HS}$ unless the dominant production processes are
$\overline{H}+H(S+S)\to\phi+\phi$ via t- and u-channels.
