  We study universal features in the shape dependence of entanglement entropy
in the vacuum state of a conformal field theory (CFT) on $\mathbb{R}^{1,d-1}$.
We consider the entanglement entropy across a deformed planar or spherical
entangling surface in terms of a perturbative expansion in the infinitesimal
shape deformation. In particular, we focus on the second order term in this
expansion, known as the entanglement density. This quantity is known to be
non-positive by the strong-subadditivity property. We show from a purely field
theory calculation that the non-local part of the entanglement density in any
CFT is universal, and proportional to the coefficient $C_T$ appearing in the
two-point function of stress tensors in that CFT. As applications of our
result, we prove the conjectured universality of the corner term coefficient
$\frac{\sigma}{C_T}=\frac{\pi^2}{24}$ in $d=3$ CFTs, and the holographic Mezei
formula for entanglement entropy across deformed spheres.
