  We investigate the cosmological behavior in a universe governed by time
asymmetric extensions of general relativity, which is a novel modified gravity
based on the addition of new, time-asymmetric, terms on the Hamiltonian
framework, in a way that the algebra of constraints and local physics remain
unchanged. Nevertheless, at cosmological scales these new terms can have
significant effects that can alter the universe evolution, both at early and
late times, and the freedom in the choice of the involved modification function
makes the scenario able to produce a huge class of cosmological behaviors. For
basic ansatzes of modification, we perform a detailed dynamical analysis,
extracting the stable late-time solutions. Amongst others, we find that the
universe can result in dark-energy dominated, accelerating solutions, even in
the absence of an explicit cosmological constant, in which the dark energy can
be quintessence-like, phantom-like, or behave as an effective cosmological
constant. Moreover, it can result to matter-domination, or to a Big Rip, or
experience the sequence from matter to dark energy domination. Additionally, in
the case of closed curvature, the universe may experience a cosmological bounce
or turnaround, or even cyclic behavior. Finally, these scenarios can easily
satisfy the observational and phenomenological requirements. Hence, time
asymmetric cosmology can be a good candidate for the description of the
universe.
