  A pseudo-Nambu-Goldstone boson (pNGB) is an attractive candidate for dark
matter (DM) due to the simple evasion of the current severe limits of DM direct
detection experiments. One of the pNGB DM models has been proposed based on a
{\it gauged} $U(1)_{B-L}$ symmetry. The pNGB has long enough lifetime to be a
DM and thermal relic abundance of pNGB DM can be fit with the observed value
against the constraints on the DM decays from the cosmic-ray observations. The
pNGB DM model can be embedded into an $SO(10)$ pNGB DM model in the framework
of an $SO(10)$ grand unified theory, whose $SO(10)$ is broken to the Pati-Salam
gauge group at the unified scale, and further to the Standard Model gauge group
at the intermediate scale. Unlike the previous pNGB DM model, the parameters
such as the gauge coupling constants of $U(1)_{B-L}$, the kinetic mixing
parameter of between $U(1)_Y$ and $U(1)_{B-L}$ are determined by solving the
renormalization group equations for gauge coupling constants with appropriate
matching conditions. From the constraints of the DM lifetime and gamma-ray
observations, the pNGB DM mass must be less than $\mathcal{O}(100)$$\,$GeV. We
find that the thermal relic abundance can be consistent with all the
constraints when the DM mass is close to half of the CP even Higg masses.
