  We examine the stability and the strong cosmic censorship in the
Reissner-Nordstrom-de Sitter (RN-dS) black hole by investigating the evolution
of a scalar field non-minimally coupled to the curvature. We find that when the
coupling parameter is negative, the RN-dS black hole experiences instability.
The instability disappears when the coupling parameter becomes non-negative.
With the increase of the coupling parameter, the violation of the strong cosmic
censorship occurs at a larger critical charge ratio. But such an increase of
the critical charge is suppressed by the increase of the cosmological constant.
Different from the minimal coupling situation, it is possible to accommodate
$\beta\ge1$ in the near extremal black hole when the scalar field is
non-minimally coupled to curvature. The increase of the cosmological constant
can allow $\beta\ge1$ to be satisfied for even smaller value of the coupling
parameter. The existence of $\beta\ge1$ implies that the resulting curvature
can continuously cross the Cauchy horizon.
