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An ideal Fermi gas at T=0 has all states filled up to the Fermi energy E_F, which depends on particle density as E_F ∝ (n)2/3. At finite T, excitations near the Fermi surface contribute to heat capacity as C_V ∝ T, much smaller than the classical equipartition value. Pressure and other thermodynamic quantities follow from the density of states.
From Fermi-Dirac statistics, you know the average occupation of a single-particle state with energy ε is ⟨n_ε⟩ = 1/(e(ε − μ)/kT + 1). At T = 0, this step function is exactly 1 for ε < μ and exactly 0 for ε > μ. The ideal Fermi gas applies this to N non-interacting fermions confined to a box, asking: what is the ground state, and how does the system behave at low temperature?
At T = 0, the ground state is the Fermi sea: fill every single-particle state in increasing energy order, one fermion per state (respecting the Pauli exclusion principle), until all N fermions are placed. The energy of the last filled state is the Fermi energy, E_F = (ℏ²/2m)(3π²n)2/3, where n = N/V is the number density. For electrons in a typical metal, n ~ 10²⁸ m⁻³, giving E_F ~ 5 eV — equivalent to a temperature of roughly 60,000 K. Even at absolute zero, the electrons have enormous kinetic energy, and the zero-temperature pressure (the degeneracy pressure) does not vanish. This quantum pressure supports white dwarf stars against gravitational collapse.
The low-temperature behavior reveals another dramatic departure from classical intuition. Classically, each gas particle contributes (3/2)k to the heat capacity, giving C_V = (3/2)Nk = (3/2)R per mole. But in a Fermi gas at temperature T, only electrons within approximately kT of the Fermi surface can be thermally excited — those deep in the Fermi sea cannot jump upward because all nearby states are already occupied. The fraction of electrons that participate is roughly kT/E_F, so the heat capacity is reduced by this factor: C_V ≈ (π²/2)(kT/E_F)Nk ∝ T, linear in temperature rather than constant. For typical metals at room temperature, kT/E_F ~ 300 K / 60,000 K ~ 0.005, so the electronic heat capacity is about 200 times smaller than the classical prediction. This explains why the heat capacity of metals is dominated by lattice vibrations (phonons ∝ T³) at low T and rises only weakly, with electronic contributions showing up as the linear term in careful measurements at very low temperatures — a key confirmation that conduction electrons behave as a degenerate Fermi gas.