  We discuss anharmonicity of the multi-octupole-phonon states in $^{208}$Pb
based on a covariant density functional theory, by fully taking into account
the interplay between the quadrupole and the octupole degrees of freedom. Our
results indicate the existence of a large anharmonicity in the transition
strengths, even though the excitation energies are similar to those in the
harmonic limit. We also show that the quadrupole-shape fluctuation
significantly enhances the fragmentation of the two-octupole-phonon states in
$^{208}$Pb. Using those transition strengths as inputs to coupled channels
calculations, we then discuss the fusion reaction of $^{16}$O+$^{208}$Pb at
energies around the Coulomb barrier. We show that the anharmonicity of the
octupole vibrational excitation considerably improves previous coupled-channels
calculations in the harmonic oscillator limit, significantly reducing the
height of the main peak in the fusion barrier distribution.
