  We study entanglement in states of holographic CFTs defined by Euclidean path
integrals over geometries with slowly varying metrics. In particular, our CFT
spacetimes have $S^1$ fibers whose size $b$ varies along one direction ($x$) of
an ${\mathbb R}^{d-1}$ base. Such examples respect an ${\mathbb R}^{d-2}$
Euclidean symmetry. Treating the $S^1$ direction as time leads to a thermofield
double state on a spacetime with adiabatically varying redshift, while treating
another direction as time leads to a confining ground state with slowly varying
confinement scale. In both contexts the entropy of slab-shaped regions defined
by $|x - x_0| \le L$ exhibits well-known phase transitions at length scales $L=
L_{crit}$ characterizing the CFT entanglements. For the thermofield double, the
numerical coefficients governing the effect of variations in $b(x)$ on the
transition are surprisingly small and exhibit an interesting change of sign:
gradients reduce $L_{crit}$ for $d \le 3$ but increase $L_{crit}$ for $d\ge4$.
This means that, while for general $L > L_{crit}$ they significantly increase
the mutual information of opposing slabs as one would expect, for $d\ge 4$
gradients cause a small decrease near the phase transition. In contrast, for
the confining ground states gradients always decrease $L_{crit}$, with the
effect becoming more pronounced in higher dimensions.
