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It entirely depends on what that electron pair is doing.
If the lone pair is hanging out by itself in its own orbital, then the carbon is probably sp3 hybridized, as greater hybridization is typically more stable. This is what you'll see on an alcohol or an amine lone pair. Those lone pairs aren't really interacting with anything else in the molecule, hence the "standard" logic of the rule you invoke.
However, in certain compounds you're able to stabilize the system further by delocalizing the electron pair. In this sort of case, the sp3 hybridized lone pair orbital is not able to participate in the delocalization, and thus will be higher in energy than an electron pair that is in the remaining p orbital of an sp2 hybridized carbon. This is the sort of thing you see with amide groups, where the "lone pair" on the nitrogen is sp2-like, which allows the electron pair to delocalize into the carbonyl, allowing for resonance stabilization.
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