  In this paper we investigate the cosmological dynamics of an up to cubic
curvature correction to General Relativity (GR) known as Cosmological
Einsteinian Cubic Gravity (CECG), whose vacuum spectrum consists of the
graviton exclusively and its cosmology is well-posed as an initial value
problem. We are able to uncover the global asymptotic structure of the phase
space of this theory. It is revealed that an inflationary matter-dominated
bigbang is the global past attractor which means that inflation is the starting
point of any physically meaningful cosmic history. Given that higher order
curvature corrections to GR are assumed to influence the cosmological dynamics
at early times -- high energies/large curvature limit -- the late-time
inflation can not be a consequence of the up to cubic order curvature
modifications. We confirm this assumption by showing that late-time
acceleration of the expansion in the CECG model is possible only if add a
cosmological constant term.
