  We consider the nonlinear damped Klein-Gordon equation \[
\partial_{tt}u+2\alpha\partial_{t}u-\Delta u+u-|u|^{p-1}u=0 \quad \text{on} \ \
[0,\infty)\times \mathbb{R}^N \] with $\alpha>0$, $2 \le N\le 5$ and energy
subcritical exponents $p>2$. We study the behavior of solutions for which it is
supposed that only one nonlinear object appears asymptotically for large times,
at least for a sequence of times.
  We first prove that the nonlinear object is necessarily a bound state. Next,
we show that when the nonlinear object is a non-degenerate state or a
degenerate excited state satisfying a simplicity condition, the convergence
holds for all positive times, with an exponential or algebraic rate
respectively. Last, we provide an example where the solution converges exactly
at the rate $t^{-1}$ to the excited state.
