We always draw f(E) vs. E crossing at $[E_F, 0.5]$ for any temperature. But a new temperature is a different steady state. So why the value of $E_F$ (Fermi level) doesn't change with temperature?
No actual reason was really given here.
Answer
I'm not quite sure what you mean by "crossing at $[E_F,0.5]$" but the Fermi level doesn't change with temperature by definition. The Fermi level is defined as the energy of the highest energy electrons at zero temperature when the system is in its ground state. It is a property of the system that is only dependent on the quantum mechanical eigenfunctions and the number of electrons and is independent of thermodynamics or statistical mechanics. When you increase the temperature of the system, you can excite electrons above the Fermi level.
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