= Solution
For a <phonon> mode of angular frequency $\omega$, the equilibrium occupation is
$$
n_\omega=\frac1{e^{\hbar\omega/(k_BT)}-1}.
$$
Well above the <Debye temperature>, the relevant vibrational <energies> are small compared with $k_BT$. Expanding the exponential gives
$$
n_\omega\simeq\frac{k_BT}{\hbar\omega},\qquad
N_{\mathrm{ph}}=\sum_{\mathbf q,s}n_{\mathbf q,s}
\simeq\frac{k_BT}{\hbar}\sum_{\mathbf q,s}\frac1{\omega_{\mathbf q,s}}.
$$
The spectrum and its cutoff are fixed in the model, so the remaining sum is <temperature> independent. Thus
$$
\boxed{N_{\mathrm{ph}}\propto T,\qquad\rho\propto T\quad(T\gg\Theta_D).}
$$
For example, the three-dimensional <Debye model> with $D(\omega)=9N\omega^2/\omega_D^3$ gives $N_{\mathrm{ph}}\simeq(9/2)NT/\Theta_D$. This is the <high-temperature phonon number in the Debye model>. The number of <phonon> quanta can exceed $3N$; $3N$ counts modes, not their total occupation. Under the stipulated proportionality to <phonon> number, only the <phonon> contribution to resistivity has been calculated. A temperature-independent <residual resistivity> would add an intercept in a real sample.
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