  In this paper we study the r-stability of closed spacelike hypersurfaces with
constant $r$-th mean curvature in conformally stationary spacetimes of constant
sectional curvature. In this setting, we obtain a characterization of
$r-$stability through the analysis of the first eigenvalue of an operator
naturally attached to the $r$-th mean curvature. As an application, we treat
the case in which the spacetime is the de Sitter space.
