  In this paper, we construct a viable model for a GeV scale self-interacting
dark matter (DM), where the DM was thermally produced in the early universe.
Here, a new vector-like fermion with a dark charge under the $U(1)_{D}$ gauge
symmetry serves as a secluded WIMP DM and it can dominantly annihilate into the
light dark gauge boson and singlet scalar through the dark gauge interaction.
Also, the self-interaction of DM is induced by the light dark gauge boson via
the same gauge interaction. In addition to these particles, we further
introduce two Weyl fermions and a doublet scalar, by which the dark gauge boson
produced from s-wave DM annihilations can mostly decay into active neutrinos
after the dark symmetry breaking such that the CMB bound on the DM with low
masses can be eluded. In order to have a common parameter region to explain the
observed relic abundance and self-interaction of DM, we also study this model
in a non-standard cosmological evolution, where the cosmic expansion driven by
a new field species is faster than the standard radiation-dominated universe
during the frozen time of DM. Reversely, one can also use the self-interacting
nature of light thermal DM to examine the non-standard cosmological history of
the universe.
