Novel black-hole solutions in Einstein-scalar-Gauss-Bonnet theories with a cosmological constant

We consider the Einstein-scalar-Gauss-Bonnet theory in the presence of a cosmological constant Λ, either positive or negative, and look for novel, regular black-hole solutions with a nontrivial scalar hair. We first perform an analytic study in the near-horizon asymptotic regime and demonstrate that a regular black-hole horizon with a nontrivial hair may always be formed, for either sign of Λ and for arbitrary choices of the coupling function between the scalar field and the Gauss-Bonnet term. At the faraway regime, the sign of Λ determines the form of the asymptotic gravitational background leading to either a Schwarzschild--anti-de Sitter--type background (Λ<0) or a regular cosmological horizon (Λ>0), with a nontrivial scalar field in both cases. We demonstrate that families of novel black-hole solutions with scalar hair emerge for Λ<0, for every choice of the coupling function between the scalar field and the Gauss-Bonnet term, whereas for Λ>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 behavior of the scalar field.