Inelastic neutron scattering was performed on silicon powder to measure the phonon density of states (DOS) from 100 to 1500 K. The mean fractional energy shifts with temperature of the modes were $$〈${}$Δ${}{$ɛ${}}_{i}/{$ɛ${}}_{i}$Δ${}T$〉${}=$-${}0.07$, giving a mean isobaric Gr\"uneisen parameter of $+6.95\ifmmode±\else\textpm\fi{}0.67$, which is significantly different from the isothermal parameter of +0.98. These large effects are beyond the predictions from quasiharmonic models using density functional theory or experimental data, demonstrating large effects from phonon anharmonicity. At 1500 K the anharmonicity contributes $0.15{k}_{\mathrm{B}}$/atom to the vibrational entropy, compared to $0.03{k}_{\mathrm{B}}$/atom from quasiharmonicity. Excellent agreement was found between the entropy from phonon DOS measurements and the reference NIST-JANAF thermodynamic entropy from calorimetric measurements.