Cosmology is then not simply an application of a relativistic version of MOND but a unit with it. The key to finding the underlying theory may lie in understanding first how an acceleration of cosmological significance can, at all, enter local dynamics, which I discuss in the last section.
If a0 is a fingerprint of cosmology on local dynamics, it is not necessarily the identification a0 ~ aex cH0 which is the the right one. There are other cosmological acceleration scales    such as ac c2 / Rc, where Rc is the curvature radius (spatial or space-time), or a c 1/2, where is the cosmological constant. Today we have only upper limits on ac, which is of the order of aex. Several pieces of evidence seem now to imply a non-zero cosmological-constant with ~ H02. If this is true then we also have a0 ~ a. Thus a0 might be a proxy for any of the cosmological acceleration parameters. Since these depend differently on cosmic time, a0 may vary with cosmic time in a way that is difficult to know without the correct identification. Such possible variation of a0 has obvious ramifications for the formation and the ensuing evolution of galactic systems.
Even without a theory we can make out some semi-quantitative aspects by which MOND cosmology must differ greatly from standard cosmology:
1. MOND is based on the phenomenology of galactic systems and hence, in principle, is not committed on the question of cosmologically homogeneous component of dark matter. But certainly, it is in the spirit of MOND that we should not conjecture the existence of any DM component without first trying to explain it away with new physics. Recent leanings toward a non-zero cosmological constant(CC) are a step in this direction. And perhaps  the same mechanism that produces a CC-like contribution might also effect MOND (hence the coincidence a0 ~ c 1/2). At any event, a MOND-inspired cosmology would start with no dark matter.
2. The MOND Jeans mass-a basic concept in structure formation, which indicates which masses are likely to collapse from an homogeneous medium-depends differently on the temperature, T, and density, , of the medium: MJ(MOND) T2 / a0, instead of the Newtonian dependence MJ T3/2 -1/2.
3. The acceleration in a collapsing system increases as the collapse proceeds (after detachment from the Hubble flow). If a0 varies at all, it is expected to decrease with cosmic time. So, the effect of MOND is expected to decrease with time in a collapsing system. (The system would behave as if the fraction of fictitious dark matter it harbors decreases with time.)
In default of a theory one can still attempt to obtain approximate MOND cosmologies-in order to get a hint of what is expected-by supplementing nonrelativistic MOND with extra assumptions. For instance, one might assume that a0 does not vary with cosmic time, identifying it with a veritable cosmological constant . This is done in ref.  where some further tentative assumptions are made. In such a case one is bound to ask why it is that this constant a0 is today of the same order as the variable cH0. The same question arises in connection with the emerging value of the cosmological constant ~ H02. In MOND, at any rate, this could find an antropic explanation whereby structure formation (hence star formation and the eventual development of mankind) is facilitated when the acceleration within the horizon (~ cH0) - decreasing as it does with cosmic time - becomes similar to the crucial dynamical constant a0  .