Is there any relation between temperature dependence of resistance and fermi energy in metals? Given that the resistance varies linearly with temperature in metals, is there any way we can calculate the Fermi energy from this information?
 A: The fact that the conductivity $\sigma=\frac{1}{\rho}$ of a metal scales like $\propto\frac{1}{T}$ is due to elastic electron-phonon scattering, i.e. the coulombian interaction between the charge density fluctuation induce by a phonon and an electron.
In an incoherent transport theory of electron in solids (i.e. avoiding corrections due to interferences like weak localization), the conductivity is well described by the Drude model :
$$
\sigma=\frac{ne^2\tau}{m}
$$
where $\tau$ is the typical time scale for the electron to "collide" and change its momentum, $n$ is the electron density. There are various processes that determine $\tau$, one is elastic electron-phonon scattering, but there is inelastic electron-phonon scattering, electron-electron scattering and scattering on static impurities too, etc. All of these processes have different scaling in respect of the temperature $T$.
It turns out that at hight temperature (room temperature), elastic electron-phonon scattering is the dominant one. It can be shown that :
$$
\tau_{\text{el-ph}}=\frac{\hbar}{k_B T}
$$
which does not depend on the Fermi energy. However, the Fermi momentum $k_F$ and the electron density $n$ are bound to each other by :
$$
k_F^3=3\pi^2n\quad\text{i.e.}\quad n=\frac{1}{3\pi^2}\left(\frac{2mE_F}{\hbar^2}\right)^{3/2}
$$
Then the room temperature resistivity reads :
$$
\rho=\frac{3\pi^2m\,k_BT}{\hbar e^2}\left(\frac{\hbar^2}{2mE_F}\right)^{3/2}\propto T
$$
A: In my point of view, it is independent thing.
Dependence of resistance on temperature is determined by Nernst-Einstein equation.
$$
R=\frac{l k_B T}{S D Z^2 e^2 C}
$$
where $T$ -is a temperature of resistance, $k_B$ is a Boltzmann constant, l- length, S - cross sectional area, $D$ - diffusion coefficient, $C$ is a charge carrier density, $Z$ is a amount of electric charge carrier, and $e$ is a electron charge.
This equation is followed from kinetic theory. 
In point of view Fermi energy is required than there is not band gap(it means that it is metal). 
