  The holographic entanglement entropy functional for higher-curvature
gravities involves a weighted sum whose evaluation, beyond quadratic order,
requires a complicated theory-dependent splitting of the Riemann tensor
components. Using the splittings of general relativity one can obtain
unambiguous formulas perturbatively valid for general higher-curvature
gravities. Within this setup, we perform a novel rewriting of the functional
which gets rid of the weighted sum. The formula is particularly neat for
general cubic and quartic theories, and we use it to explicitly evaluate the
corresponding functionals. In the case of Lovelock theories, we find that the
anomaly term can be written in terms of the exponential of a differential
operator. We also show that order-$n$ densities involving $n_R$ Riemann tensors
(combined with $n-n_R$ Ricci's) give rise to terms with up to $2n_R-2$
extrinsic curvatures. In particular, densities built from arbitrary Ricci
curvatures combined with zero or one Riemann tensors have no anomaly term in
their functionals. Finally, we apply our results for cubic gravities to the
evaluation of universal terms coming from various symmetric regions in general
dimensions. In particular, we show that the universal function characteristic
of corner regions in $d=3$ gets modified in its functional dependence on the
opening angle with respect to the Einstein gravity result.
