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In this paper, we construct a viable model for a GeV scale self-interactingdark matter (DM), where the DM was thermally produced in the early universe.Here, a new vector-like fermion with a dark charge under the $U(1)_{D}$ gaugesymmetry serves as a secluded WIMP DM and it can dominantly annihilate into thelight 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 viathe same gauge interaction. In addition to these particles, we furtherintroduce two Weyl fermions and a doublet scalar, by which the dark gauge bosonproduced from s-wave DM annihilations can mostly decay into active neutrinosafter the dark symmetry breaking such that the CMB bound on the DM with lowmasses can be eluded. In order to have a common parameter region to explain theobserved relic abundance and self-interaction of DM, we also study this modelin a non-standard cosmological evolution, where the cosmic expansion driven bya new field species is faster than the standard radiation-dominated universeduring the frozen time of DM. Reversely, one can also use the self-interactingnature of light thermal DM to examine the non-standard cosmological history ofthe universe.

Light Thermal Self-Interacting Dark Matter in the Shadow of Non-Standard Cosmology