  According to theoretical physics the cosmological constant (CC) is expected
to be much larger in magnitude than other energy densities in the universe,
which is in stark contrast to the observed Big Bang evolution. We address this
old CC problem not by introducing an extremely fine-tuned counterterm, but in
the context of modified gravity in the Palatini formalism. In our model the
large CC term is filtered out, and it does not prevent a standard cosmological
evolution. We discuss the filter effect in the epochs of radiation and matter
domination as well as in the asymptotic de Sitter future. The final expansion
rate can be much lower than inferred from the large CC without using a
fine-tuned counterterm. Finally, we show that the CC filter works also in the
Kottler (Schwarzschild-de Sitter) metric describing a black hole environment
with a CC compatible to the future de Sitter cosmos.
