\langle E \rangle=\frac{1}{2}\hbar\omega_0+\frac{\hbar\omega_0}{ {\rm e}^{\hbar\omega_0/k_{\rm B}T}-1}
$$
The left plot below shows the Bose-Einstein distribution vs energy. We see that low-energy states are more likely to be occupied than high-energy states. The right plot shows the increasing thermal energy in the oscillator for increasing temperature and highlights the zero-point energy $\hbar\omega_0/2$ that remains in the oscillator at $T=0$ because of the uncertainty principle. Moreover, we see that the energy in the oscillator only starts to increase significantly when $kT>\hbar \omega$. I.e., the heat capacity only becomes significant for $kT>\hbar \omega$ and goes to zero when $T\rightarrow0$
The left plot below shows the Bose-Einstein distribution vs energy. We see that low-energy states are more likely to be occupied than high-energy states. The right plot shows the increasing thermal energy in the oscillator for increasing temperature and highlights the zero-point energy $\hbar\omega_0/2$ that remains in the oscillator at $T=0$ (a consequence of the uncertainty principle). Moreover, we see that the energy in the oscillator is approximately constant when $kT<\hbar \omega_0$. I.e., the heat capacity becomes small for $kT<\hbar \omega_0$ and goes to zero when $T\rightarrow0$.
