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We study entanglement in states of holographic CFTs defined by Euclidean pathintegrals over geometries with slowly varying metrics. In particular, our CFTspacetimes have $S^1$ fibers whose size $b$ varies along one direction ($x$) ofan ${\mathbb R}^{d-1}$ base. Such examples respect an ${\mathbb R}^{d-2}$Euclidean symmetry. Treating the $S^1$ direction as time leads to a thermofielddouble state on a spacetime with adiabatically varying redshift, while treatinganother direction as time leads to a confining ground state with slowly varyingconfinement scale. In both contexts the entropy of slab-shaped regions definedby $|x - x_0| \le L$ exhibits well-known phase transitions at length scales $L=L_{crit}$ characterizing the CFT entanglements. For the thermofield double, thenumerical coefficients governing the effect of variations in $b(x)$ on thetransition 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 increasethe mutual information of opposing slabs as one would expect, for $d\ge 4$gradients cause a small decrease near the phase transition. In contrast, forthe confining ground states gradients always decrease $L_{crit}$, with theeffect becoming more pronounced in higher dimensions.

Adiabatic corrections to holographic entanglement in thermofield doubles and confining ground states