We present the analysis of linear stability and causality of the third order relativistic hydrodynamics derived in the PRC 88, 021903 . Here a third order evolution equation for shear stress tensor is derived from relativistic Boltzmann equation using Chapman-Enskog expansion. We perturb the fluid system, which is initially in equilibrium and at rest, by slightly changing the energy density and fluid velocity to study its propagation. The dispersion relation for longitudinal and transverse modes of propagation is derived. It was found that there exists an acausal mode in theory around the static equilibrium. Our results match with the more detailed and elaborate analysis done in arXiv:2107.10319 [nucl-th].