In this paper, we consider the nonlinear Klein-Gordon equation
∂
t
t
u
−
Δ
u
+
u
=
|
u
|
p
−
1
u
,
t
∈
R
,
x
∈
R
d
,
\begin{align*} \partial _{tt}u-\Delta u+u=|u|^{p-1}u,\qquad t\in \mathbb {R},\ x\in \mathbb {R}^d, \end{align*}
with
1
>
p
>
1
+
4
d
1>p> 1+\frac {4}{d}
. The equation has the standing wave solutions
u
ω
=
e
i
ω
t
ϕ
ω
u_\omega =e^{i\omega t}\phi _{\omega }
with the frequency
ω
∈
(
−
1
,
1
)
\omega \in (-1,1)
, where
ϕ
ω
\phi _{\omega }
is the solution of
−
Δ
ϕ
+
(
1
−
ω
2
)
ϕ
−
ϕ
p
=
0.
\begin{align*} -\Delta \phi +(1-\omega ^2)\phi -\phi ^p=0. \end{align*}
It was proved by Shatah [Comm. Math. Phys. 91 (1983), pp. 313–327], and Shatah-Strauss [Comm. Math. Phys. 100 (1985), pp. 173–190] that there exists a critical frequency
ω
c
∈
(
0
,
1
)
\omega _c\in (0,1)
such that the standing waves solution
u
ω
u_\omega
is orbitally stable when
ω
c
>
|
ω
|
>
1
\omega _c>|\omega |>1
, and orbitally unstable when
|
ω
|
>
ω
c
|\omega |>\omega _c
. Furthermore, the strong instability for the critical frequency
|
ω
|
=
ω
c
|\omega |=\omega _c
in the high dimensions
d
≥
2
d\ge 2
was proved by Ohta-Todorova [SIAM J. Math. Anal. 38 (2007), pp. 1912–1931]. In this paper, we settle the only remaining problem when
|
ω
|
=
ω
c
|\omega |=\omega _c
,
p
>
1
p> 1
, and
d
=
1
d=1
, in which case we prove that the standing wave solution
u
ω
u_\omega
is orbitally unstable.