  We consider the Brans-Dicke (BD) theory of gravity and explore the
cosmological implications of the sign-changeable interacting holographic dark
energy (HDE) model in the background of Friedmann-Robertson-Walker (FRW)
universe. As the system's infrared (IR) cutoff, we choose the future event
horizon, the Granda-Oliveros (GO) and the Ricci cutoffs. For each cutoff, we
obtain the density parameter, the equation of state (EoS) and the deceleration
parameter of the system. In case of future event horizon, we find out that the
EoS parameter, $w_{D}$, can cross the phantom line, as a result the transition
from deceleration to acceleration expansion of the universe can be achieved
provided the model parameters are chosen suitably. Then, we investigate the
instability of the sign-changeable interacting HDE model against perturbations
in BD theory. For this purpose, we study the squared sound speed $v_s^2$ whose
sign determines the stability of the model. When $v_s^2<0$ the model is
unstable against perturbation. For future event horizon cutoff, our universe
can be stable (${v}^{2}_s>0$) depending on the model parameters. Then, we focus
on GO and Ricci cutoffs and find out that although other features of these two
cutoffs seem to be consistent with observations, they cannot leads to stable
dominated universe, except in special case with GO cutoff. Our studies confirm
that for the sign-changeable HDE model in the setup of BD cosmology, the event
horizon is the most suitable horizon which can passes all conditions and leads
to a stable DE dominated universe.
