We investigate the temperature effect on the electronic band structure and optical absorption property of wide-band-gap ternary nitride MgSiN$_2$ using first-principles calculations. We find that electron-phonon coupling leads to a giant reduction in the indirect gap of MgSiN$_2$, which is indispensable in understanding the optoelectronic properties of this material. Moreover, higher-order electron-phonon coupling terms in MgSiN$_2$ captured by the Monte Carlo calculations play an important role, especially at higher temperatures. Taking the band gap renormalization into account, the band gap of MgSiN$_2$ determined by the quasiparticle GW0 calculation shows good agreement with recent experimental result. The predicted phonon-assisted indirect optical absorption spectra show that with increasing temperature the absorption onset undergoes a red-shift and the absorption peaks become smoother. Our work provides helpful insights to the nature of the band gap of MgSiN$_2$ and facilitates its application in ultraviolet optoelectronic devices.
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