The headband of fate's favor states that it applies to any ability that gives a luck bonus or explicitly increases the good luck by increasing that effect's duration by 1 round.
The Sleut's luck power gives the sleuth a luck pool which can be used to fuel several deeds. Giving a luck pool sounds like something that explicitly increases one's luck, so would the headband's effect apply? And if so how?
Only thing it'd do for that archetype is increase
Run Like Hell (Ex): At 4th level, a sleuth can spend 1 luck point to gain a +20-foot bonus to his speed for 1 minute. Furthermore, while under the effect of this bonus, if he moves more than his normal speed in a round, he gains a +4 bonus to AC until the start of his next turn.
by one round as that's the only deed that doesn't give a "right now" bonus and is timed base.