We consider damped
s
s
-fractional Klein–Gordon equations on
R
d
\mathbb {R}^d
, where
s
s
denotes the order of the fractional Laplacian. In the one-dimensional case
d
=
1
d = 1
, Green (2020) established that the exponential decay for
s
≥
2
s \geq 2
and the polynomial decay of order
s
/
(
4
−
2
s
)
s/(4-2s)
hold if and only if the damping coefficient function satisfies the so-called geometric control condition. In this note, we show that the
o
(
1
)
o(1)
energy decay is also equivalent to these conditions in the case
d
=
1
d=1
. Furthermore, we extend this result to the higher-dimensional case: the logarithmic decay, the
o
(
1
)
o(1)
decay, and the thickness of the damping coefficient are equivalent for
s
≥
2
s \geq 2
. In addition, we also prove that the exponential decay holds for
0
>
s
>
2
0 > s > 2
if and only if the damping coefficient function has a positive lower bound, so in particular, we cannot expect the exponential decay under the geometric control condition.