  This is the second of two works, in which we discuss the definition of an
appropriate notion of mass for static metrics, in the case where the
cosmological constant is positive and the model solutions are compact. In the
first part, we have established a positive mass statement, characterising the
de Sitter solution as the only static vacuum metric with zero mass. In this
second part, we prove optimal area bounds for horizons of black hole type and
of cosmological type, corresponding to Riemannian Penrose inequalities and to
cosmological area bounds \`a la Boucher-Gibbons-Horowitz, respectively.
Building on the related rigidity statements, we also deduce a uniqueness result
for the Schwarzschild--de Sitter spacetime.
