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

Positive cosmological constant, non-local gravity and horizon entropy