We construct different neutral blackfold solutions in Anti-de Sitter and de
Sitter background spacetimes in the limit where the cosmological scale is taken
to be much larger than the transverse horizon size. This includes a class of
blackfolds with horizons that are products of odd-spheres times a transverse
sphere, for which the thermodynamic stability is also studied. Moreover, we
exhibit a specific case in which the same blackfold solution can describe
different limiting black hole spacetimes therefore illustrating the geometric
character of the blackfold approach. Furthermore, we show that the
higher-dimensional Kerr-(Anti)-de Sitter black hole allows for ultra-spinning
regimes in the relevant limit of large cosmological scale, and demonstrate that
this is correctly described by a pancaked blackfold geometry. We also give
evidence for the possibility of saturating the rigidity theorem in these
backgrounds.Comment: v1: 1+29 pages, latex. v2: cosmetic changes. v3: .pdf file correcte
