We consider a simple scalar dark matter model within the frame of gauged $L_{\mu}-L_{\tau}$ symmetry. A new gauge boson $Z'$, as well as two scalar fields $S$ and $\Phi$, are introduced to the Standard Model (SM), where $S$ and $\Phi$ are SM singlets but both carry $U(1)_{L_{\mu}-L_{\tau}}$ charge. The real component and imaginary component of $S$ can acquire different masses after spontaneously symmetry breaking, and the lighter one can play the role of dark matter which is stabilized by an extra $Z_2$ symmetry. We consider the possibility of light dark matter as well as the co-annihilation case in our model, and present current $(g-2)_{\mu}$ anomaly, Higgs invisible decay, dark matter relic density as well as direct detection constraints on the parameter space. Direct detection gives the most stringent restriction on the model and we found although most of the parameter space are excluded, a few GeV mass region is still allowed where the co-annihilation process is also permitted in the model. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Science and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd.