  In this paper we investigate the non-Abelian cosmic string in de Sitter and
anti-de Sitter spacetimes. In order to do that we construct the complete set of
equations of motion considering the presence of a cosmological constant. By
using numerical analysis we provide the behavior of the Higgs and gauge fields
and also for the metric tensor for specific values of the physical parameters
of the theory. For de Sitter case, we find the appearance of horizons that
although being consequence of the presence of the cosmological constant it
strongly depends on the value of the gravitational coupling. In the anti-de
Sitter case, we find that the system does not present horizons. In fact the new
feature of this system is related with the behavior of the $(00)$ and $(zz)$
components of the metric tensor. They present a strongly increasing for large
distance from the string.
