Motivated by the growing evidence for the possible lepton flavour universality violation after the first results from Fermilabs muon $(g-2)$ measurement, we revisit one of the most widely studied anomaly free extensions of the standard model namely, gauged $L_{mu}-L_{tau}$ model, to find a common explanation for muon $(g-2)$ as well as baryon asymmetry of the universe via leptogenesis. The minimal setup allows TeV scale resonant leptogenesis satisfying light neutrino data while the existence of light $L_{mu}-L_{tau}$ gauge boson affects the scale of leptogenesis as the right handed neutrinos are charged under it. For $L_{mu}-L_{tau}$ gauge boson mass at GeV scale or above, the muon $(g-2)$ favoured parameter space is already ruled out by other experimental data while bringing down its mass to sub-GeV regime leads to vanishing lepton asymmetry due to highly restrictive structures of lepton mass matrices at the scale of leptogenesis. Extending the minimal model with two additional Higgs doublets can lead to a scenario consistent with successful resonant leptogenesis and muon $(g-2)$ while satisfying all relevant experimental data.
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