  We consider the Einstein-scalar-Gauss-Bonnet theory in the presence of a
cosmological constant $\Lambda$, either positive or negative, and look for
novel, regular black-hole solutions with a non-trivial scalar hair. We first
perform an analytic study in the near-horizon asymptotic regime, and
demonstrate that a regular black-hole horizon with a non-trivial hair may be
always formed, for either sign of $\Lambda$ and for arbitrary choices of the
coupling function between the scalar field and the Gauss-Bonnet term. At the
far-away regime, the sign of $\Lambda$ determines the form of the asymptotic
gravitational background leading either to a Schwarzschild-Anti-de Sitter-type
background ($\Lambda<0$) or a regular cosmological horizon ($\Lambda>0$), with
a non-trivial scalar field in both cases. We demonstrate that families of novel
black-hole solutions with scalar hair emerge for $\Lambda<0$, for every choice
of the coupling function between the scalar field and the Gauss-Bonnet term,
whereas for $\Lambda>0$, no such solutions may be found. In the former case, we
perform a comprehensive study of the physical properties of the solutions found
such as the temperature, entropy, horizon area and asymptotic behaviour of the
scalar field.
