We formulate a generally covariant six-field Rational Extended Thermodynamics model (RET6) for relativistic polyatomic gases, with the dynamical pressure as the only non-equilibrium variable. The model is based on a polyatomic extension of the Boltzmann–Chernikov kinetic equation, where the one-particle distribution depends also on an internal-energy variable, and on the Maximum Entropy closure of the associated relativistic moment hierarchy. The resulting field equations, closure relations, and production term are therefore fixed by the underlying kinetic structure rather than postulated phenomenologically. We extend the RET6 model from Minkowski spacetime to a general curved spacetime by the minimal coupling prescription and couple it to the Einstein equations. As a first structural result, we prove a kinetic-theory no-go theorem in this polyatomic RET setting stating that any stress-energy tensor induced by a non-negative relativistic one-particle distribution function satisfies the strong energy condition. We then specialize the theory to a homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime. In this setting the dynamical pressure modifies the expansion dynamics with respect to the perfect-fluid Euler case, but the no-go theorem excludes acceleration driven by the RET6 gas alone. Finally, we reintroduce a cosmological constant and study the combined ΛRET6 model. For the diatomic equation of state and a constant positive relaxation time, we prove the existence and local stability of a de Sitter attractor at late times. Numerical integrations show that, for representative post-recombination initial data and constant relaxation times, the expansion history rapidly approaches that of ΛCDM, with small non-equilibrium corrections controlled by the relaxation time and by the initial value of the dynamical pressure.