  We construct a class of generalized non-local gravity (GNLG) model which is
the modified theory of general relativity (GR) obtained by adding a term
$m^{2n-2} R\Box^{-n}R$ to the Einstein-Hilbert action. Concretely, we not only
study the gravitational equation for the GNLG model by introducing auxiliary
scalar fields, but also analyse the classical stability and examine the
cosmological consequences of the model for different exponent $n$. We find that
the half of the scalar fields are always ghost-like and the exponent $n$ must
be taken even number for a stable GNLG model. Meanwhile, the model
spontaneously generates three dominant phases of the evolution of the universe,
and the equation of state parameters turn out to be phantom-like. Furthermore,
we clarify in another way that exponent $n$ should be even numbers by discuss
the spherically symmetric static solutions in Newtonian gauge. It is worth
stressing that the results given by us can include ones in refs. [28, 34] as
the special case of $n=2$.
