Here's my take.
According to the intro to metamagic rods, Hyp is right, and it's applied at preparation time for prepared spellcasters. The "ability to use the feat" is pretty clear on that.
But, that's not how I voted, because all of the specific rods say something different.
They say "can cast up to three spells a day that are empowered as though using the Empower Spell feat." and so forth. They don't say that you actually use the feat AT ALL! They just say that the spell is cast in a state AS THOUGH you used the feat (which, in fact, means that you didn't use the feat.) And if all they do is apply the effect of the feat on the spell, then everyone uses them at casting time, just like it says.
This is, to me, inherently contradictory. In one place, you use the feat, and in the other place, the spell is effected as though you had used the feat. Normally, the difference would be too subtle to matter, but here, it is huge.
The only way I can reconcile this is to say that "the ability to use the feat" is a very bad way of saying "the ability to apply the effects of the feat". It's a stretch, but it's better than saying that each rod is actually saying "cast up to three spells a day which, at the appropriate time for your class, you applied the Empower Spell metamagic feat to without increasing the spell level using this item, which is also limited to three uses per day."
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gnfnrf