  We measured phonon dispersion in single crystal EuTiO$_3$ using inelastic
x-ray scattering. A structural transition to an antiferrodistortive phase was
found at a critical temperature $T_0$=287$\pm$1 K using powder and
single-crystal x-ray diffraction. Clear softening of the zone boundary
\emph{R}-point \textbf{q}=(0.5 0.5 0.5) acoustic phonon shows this to be a
displacive transition. The mode energy plotted against reduced temperature
could be seen to nearly overlap that of $\rm SrTiO_3$, suggesting a universal
scaling relation. Phonon dispersion was measured along $\Gamma$-$X$ (0 0
0)$\rightarrow$(0.5 0 0). Mode eigenvectors were obtained from a shell model
consistent with the \textbf{q}-dependence of intensity and energy, which also
showed that the dispersion is nominally the same as in $\rm SrTiO_3$ at room
temperature, but corrected for mass. The lowest energy optical mode, determined
to be of Slater character, softens approximately linearly with temperature
until the 70-100 K range where the softening stops, and at low temperature, the
mode disperses linearly near the zone center.
