  Let $u$ solve the damped Klein--Gordon equation $$ \big( \partial_t^2-\sum
\partial_{x_j}^2 +m \text{Id} +\gamma(x) \partial_t \big) u=0 $$ on
$\mathbb{R}^n$ with $m>0$ and $\gamma\geq 0$ bounded below on a $2 \pi
\mathbb{Z}^n$-invariant open set by a positive constant. We show that the
energy of the solution $u$ decays at a polynomial rate. This is proved via a
periodic observability estimate on $\mathbb{R}^n.$
