Double strangeness ${\Xi }^{-}$ production in Au+Au collisions at 2, 4, and 6 GeV/nucleon incident beam energies is studied with the pure hadron cascade version of a multi-phase transport model. It is found that due to larger nuclear compression, the model with the soft equation of state (EoS) gives larger yields of both single strangeness ($K^{+}$ and $\Lambda +{\Sigma }^0$) and double strangeness ${\Xi }^{-}$. The sensitivity of the double strangeness ${\Xi }^{-}$ to the EoS is evidently larger than that of $K^{+}$ or $\Lambda +{\Sigma }^0$ since the phase-space distribution of produced ${\Xi }^{-}$ is more compact compared to those of the single strangeness. The larger sensitivity of the yields ratio of ${\Xi }^{-}$ to the EoS from heavy and light systems is kept compared to that of the single strangeness. The study of ${\Xi }^{-}$ production in relativistic heavy-ion collisions provides an alternative for the ongoing heavy-ion collision program at facilities worldwide for identifying the EoS at high densities, which is relevant to the investigation of the phase boundary and onset of deconfinement of dense nuclear matter.