  We study an example of Grand Unified Theory (GUT), known as trinification,
which was first introduced in 1984 by S.Glashow. This model has the GUT gauge
group as $[SU(3)]^3$ with a discrete $\mathbb{Z}_3$ to ensure the couplings are
unified at the GUT scale. In this letter we consider this trinification model
in its minimal formulation and investigate its robustness in the context of
cosmology. In particular we show that for a large set of the parameter space
the model doesn't seem to provide a Dark Matter candidate compatible with
cosmological data.
