  Recently proposed nonlocal and nonperturbative late time behavior of the heat
kernel is generalized to curved spacetimes. Heat kernel trace asymptotics is
dominated by two terms one of which represents a trivial covariantization of
the flat-space result and another one is given by the Gibbons-Hawking integral
over asymptotically-flat infinity. Nonlocal terms of the effective action
generated by this asymptotics might underly long- distance modifications of the
Einstein theory motivated by the cosmological constant problem. New mechanisms
of the cosmological constant induced by infrared effects of matter and graviton
loops are briefly discussed.
