  We discuss a class of (local and non-local) theories of gravity that share
same properties: i) they admit the Einstein spacetime with arbitrary
cosmological constant as a solution; ii) the on-shell action of such a theory
vanishes and iii) any (cosmological or black hole) horizon in the Einstein
spacetime with a positive cosmological constant does not have a non-trivial
entropy. The main focus is made on a recently proposed non-local model. This
model has two phases: with a positive cosmological constant $\Lambda>0$ and
with zero $\Lambda$. The effective gravitational coupling differs essentially
in these two phases. Generalizing the previous result of Barvinsky we show that
the non-local theory in question is free of ghosts on the background of any
Einstein spacetime and that it propagates a standard spin-2 particle. Contrary
to the phase with a positive $\Lambda$, where the entropy vanishes for any type
of horizon, in an Einstein spacetime with zero cosmological constant the
horizons have the ordinary entropy proportional to the area. We conclude that,
somewhat surprisingly, the presence of any, even extremely tiny, positive
cosmological constant should be important for the proper resolution of the
entropy problem and, possibly, the information puzzle.
