It does not
As the feat says, it only works on material components, not any other kind of components:
By using a divine focus as part of casting, you can cast any spell with a material component costing the value of that divine focus (maximum 100 gp) or less without needing that component.
Foci are not components:
A focus component is a prop of some sort. Unlike a material component, a focus is not consumed when the spell is cast and can be reused. As with material components, the cost for a focus is negligible unless a price is given. Assume that focus components of negligible cost are in your spell component pouch.
A material component consists of one or more physical substances or objects that are annihilated by the spell energies in the casting process. Unless a cost is given for a material component, the cost is negligible. Don’t bother to keep track of material components with negligible cost. Assume you have all you need as long as you have your spell component pouch.
Allowing False Focus to work on focus components would open a precedent for Eschew Materials, which has a very similar wording, to also work on focus components. But of course, there are very few foci with a cost of under 1gp.
Feats like Create Reliquary could be used to easily obtain a very expensive Divine Focus that would allow you to ignore many Focus and Material components in the game. Instead of paying for the foci of your spells individually, you would build a single (very expensive) divine focus and be done with it. It is a pretty strong feat as it is already.