  The entanglement entropy of a generic $d$-dimensional conformal field theory
receives a regulator independent contribution when the entangling region
contains a (hyper)conical singularity of opening angle $\Omega$, codified in a
function $a^{(d)}(\Omega)$. In arXiv:1505.04804, we proposed that for
three-dimensional conformal field theories, the coefficient $\sigma$
characterizing the smooth surface limit of such contribution
($\Omega\rightarrow \pi$) equals the stress tensor two-point function charge
$C_{ T}$, up to a universal constant. In this paper, we prove this relation for
general three-dimensional holographic theories, and extend the result to
general dimensions. In particular, we show that a generalized coefficient
$\sigma^{ (d)}$ can be defined for (hyper)conical entangling regions in the
almost smooth surface limit, and that this coefficient is universally related
to $C_{ T}$ for general holographic theories, providing a general formula for
the ratio $\sigma^{ (d)}/C_{ T}$ in arbitrary dimensions. We conjecture that
the latter ratio is universal for general CFTs. Further, based on our recent
results in arXiv:1507.06997, we propose an extension of this relation to
general R\'enyi entropies, which we show passes several consistency checks in
$d=4$ and $d=6$.
