  Curved momentum spaces associated to the $\kappa$-deformation of the (3+1) de
Sitter and Anti-de Sitter algebras are constructed as orbits of suitable
actions of the dual Poisson-Lie group associated to the $\kappa$-deformation
with non-vanishing cosmological constant. The $\kappa$-de Sitter and
$\kappa$-Anti-de Sitter curved momentum spaces are separately analysed, and
they turn out to be, respectively, half of the (6+1)-dimensional de Sitter
space and half of a space with $SO(4,4)$ invariance. Such spaces are made of
the momenta associated to spacetime translations and the "hyperbolic" momenta
associated to boost transformations. The known $\kappa$-Poincar\'e curved
momentum space is smoothly recovered as the vanishing cosmological constant
limit from both of the constructions.
