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 ratefor 0 < < 2 and at some exponential rate when ≥ 2. Our approach is based on the asymptotic theory of 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. K E Y W O R D S damped wave equation, fractional derivative, geometric control condition M S C ( 2 0 1 0 ) 35B35, 35B40, 35G30