  We investigate the Lorentz symmetry breaking effects (LSBE) on the deflection
of light by a rotating cosmic string spacetime in the weak limit approximation.
We first calculate the deflection angle by a static cosmic string for a fixed
spacelike 4-vector case (FSL) with the corresponding effective-string optical
metric using the Gauss-Bonnet theorem (GBT). Then, we focus on a more general
scenario, namely we calculate the deflection angle by a rotating cosmic string
applying the GBT to Randers effective-string metric. We obtain a significant
modification in the deflection angle because of the LSBE parameter. We find a
first and second order correction terms due to the global effective topology
which are proportional to the cosmic string and LSBE parameter, respectively.
Finally, for a fixed time-like 4-vector (FTL) case, we show that the deflection
angle is not affetced by LSBE parameter.
