  We consider the fractional Klein-Gordon equation in one spatial dimension,
subjected to a damping coefficient, which is non-trivial and periodic, or more
generally strictly positive on a periodic set. We show that the energy of the
solution decays at the polynomial rate $O(t^{-\frac{s}{4-2s}})$ for $0< s<2 $
and at some exponential rate when $s\geq 2$. Our approach is based on the
asymptotic theory of $C_0$ semigroups in which one can relate the decay rate of
the energy in terms of the resolvent growth of the semigroup generator. The
main technical result is a new observability estimate for the fractional
Laplacian, which may be of independent interest.
