You’re right: it’s more about doing. “Han har annat att tänka på” is about the mind.
Update: I can’t find this Swedish sentence on Tatoeba, but there is “Tom har annat för sig”, which is a translation of “Tom has plans.” That is also a rather free interpretation.
Many apologies - when i was messing around with machine translators i changed Tom to han. Sometimes they translate Tom as empty, so I have to change it.