My lecture derived the expression for this contribtuin using the collision integral approach but I missed lot of details. He considers the lowest order correction to distribution function $f=f_0+\delta f_p$, where $f_0$ is the equilibrium function and "guesses" the expression for $\delta f=(-\partial f_0/\partial\epsilon)T\chi$. Then he writes down the expression of collision integrals for phonon-electron interaction. After several approximations, he represents collision integral as $-\hat{\Omega}\chi_p$ where $\hat{\Omega}$ plays the role of operator in "Hilbert space". After that, he expands $\chi_p$ by a basis $\lbrace\phi_l(p)\rbrace$ and then consider only the one function $\phi(p)$ in a basis.
Of course, I lost lot of details but the key idea is to reduce the problem to "eigenvalue" problem and solve it. For me, it seems very unnatural.
Hence, I will be very grateful for how to obtain the expression for $\rho$ wich has the correct behavior $\sim T$ for high temperatures and $\sim T^5$ for low temperatures.