  We shall introduce the approximate representability and the Rohlin property
for coactions of a finite dimensional $C^*$-Hopf algebra on a unital
$C^*$-algebra and discuss some basic properties of approximately representable
coactions and coactions with the Rohlin property of a finite dimensional
$C^*$-Hopf algebra on a unital $C^*$-algebra. Also, we shall give an example of
an approximately representable coaction of a finite dimensional $C^*$-Hopf
algebra on a simple unital $C^*$-algebra which has also the Rohlin property and
we shall give the 1-cohomology vanishing theorem for coactions of a finite
dimensional $C^*$-Hopf algebra on a unital $C^*$-algebra and the 2-cohomology
vanishing theorem for twisted coactions of a finite dimensional $C^*$-Hopf
algebra on a unital $C^*$-algebra. Furthermore, we shall introduce the notion
of the approximately unitary equivalence of coactions of a finite dimensional
$C^*$-Hopf algebra $H$ on a unital $C^*$-algebra $A$ and show that if $\rho$
and $\sigma$, coactions of $H$ on a separable unital $C^*$-algebra $A$, which
have the Rohlin property, are approximately unitarily equivalent, then there is
an approximately inner automorphism $\alpha$ on $A$ such that
$\sigma=(\alpha\otimes\id)\circ\rho\circ\alpha^{-1}$.
