We have formulated the invariant ion-acoustic waves (IAWs) in the astrophysical plasmas including the pure thermodynamic features of the background particles. Here, we have used the modern version of the kappa distribution formalism, where it is labeled with an invariant kappa index as of the zero dimensionality spectral index, κ0. Two contexts for studying the invariant IAWs have been employed, i.e., the kinetic theory formalism and the hydrodynamic fluid description. At first, we have employed the Vlasov-Poisson equations at the low frequency band of the weakly damped ion waves, where the most generalized formalism of the ion-sound speed has been confirmed in terms of the extended polytropic indices of the plasma species, γj. Furthermore, the Landau damping of IAWs has been formulated in terms of κ0, the wavelength, and the temperatures of the plasma species. In the hydrodynamic description, we have normalized the fluid parameters by using the extended quantities, including the generalized ion-sound speed and Debye length. Then, by using the perturbation theory in the linear and nonlinear regimes, some missing issues in the formulation of the invariant IAWs have been emphasized, such as the effect of the perturbed potential degrees of freedom, and the combined effects of the wave steepening and dispersion. The second feature has been analyzed by deriving a generalized KdV equation and its solitary wave solutions in an invariant formalism. Here, farther/near-equilibrium regions, corresponding to the thermodynamic evolutions, have been characterized by 0 < κ0 < 1 (0 < γj < 0.5) and κ0 > 1 (0.5 < γj < 1), respectively, where we have analyzed the IAWs in the anti-equilibrium state at κ0 → 0 (γj → 0) towards the equilibrium state at κ0 ≫ 1 (γj → 1). Our numerical analysis confirms the distinction between the IAWs diagrams in two regions, where the transition from far-equilibrium states to the near-equilibrium states occurs in the vicinity κ0 ∼ 1 (γj ∼ 0.5), denoting the escape state of the evolution.