The role of temperature in Nernst equation
The standard cell potential is temperature-dependent itself, just like the Gibbs free energy. The easiest way to see this is to write the Gibbs free energy in terms of enthalpy, entropy and temperature:

$$\Delta G^\circ = \Delta H^\circ - T \Delta S^\circ$$

This shows you that $\Delta G^\circ$ is strongly temperature dependent (even if $ \Delta H^\circ$ and $\Delta S^\circ$ were independent of temperature, which they are not).

To relate this to the cell potential, remember that $\Delta G^\circ$ is a measure of the maximal work a system can do, and that the electrical work is electrical potential times charge moved across that potential.

$$\Delta G^\circ = w_\text{electrical} = -E^\circ n F$$

So $E^\circ$ is temperature dependent as well.