  We investigate the thermodynamical and causal consistency of cosmological
models of the cubic Quasi-Topological Gravity (QTG) in four dimensions, as well
as their phenomenological consequences. Specific restrictions on the maximal
values of the matter densities are derived by requiring the apparent horizon's
entropy to be a non-negative, non-decreasing function of time. The QTG
counterpart of the Einstein-Hilbert (EH) gravity model of linear equation of
state is studied in detail. An important feature of this particular QTG
cosmological model is the new early-time acceleration period of the evolution
of the Universe, together with the standard late-time acceleration present in
the original EH model. The QTG correction to the causal diamond's volume is
also calculated.
