You can probably do this, but it wouldn't work as you suggest
Channel Divinity: Knowledge of the Ages
...As an action, you choose one skill or tool. For 10 minutes, you have proficiency with the chosen skill or tool.
Yes you can choose a temporarily proficient skill as long as your DM works with you
Seeing that Knowledge of the Ages makes you proficient in a skill temporarily, it seems that there is nothing preventing it from being a valid target for Expertise. It is probably not an intended use of it, but nothing in the rules prevents it.
However, it is going to take a carefully or luckily timed use of the Knowledge of the Ages ability coinciding with the moment of level-up for this to even get off the ground. This is going to probably involve some level of meta-gaming on the part of the player (and cooperation from the DM) (or some luck) to work. Whether this is frowned upon varies from table to table though and is certainly not inherently wrong.
So, as long as the DM is fine with this, there is no rules reason why you could not choose the Knowledge of Ages skill for the target of Expertise.
If you are not proficient in a skill you gain no benefit from Expertise
Expertise does not grant proficiency, it only doubles the bonus of skill that you are already proficient in, when you make an ability check with it.
Expertise [C]hoose two of your skill proficiencies. Your proficiency bonus is doubled for any ability check you make that uses either of the chosen proficiencies.
Notice that Expertise applies only to proficiencies not skills in general. If the skill you are trying to apply it to isn't a proficiency for your character then Expertise cannot apply and nothing is doubled.
This is backed up by the PHB:
[I]f a feature or effect allows you to multiply your proficiency bonus when making an ability check that wouldn’t normally benefit from your proficiency bonus, you still don’t add the bonus to the check. For that check your proficiency bonus is 0, given the fact that multiplying 0 by any number is still 0. For instance, if you lack proficiency in the History skill, you gain no benefit from a feature that lets you double your proficiency bonus when you make Intelligence (History) checks. (PHB 174)
Thus, if you are not proficient in a skill, that skill can gain no benefit from Expertise at all. Even if you are getting a bonus from Jack of All Trades, that skill is still not a proficiency for you and thus cannot be a target of Expertise in the first place.
This is how is plays out in each case:
While Knowledge of Ages is active
While you are still proficient in the skill you would add your proficiency bonus to it. Thus Jack of All Trades would not apply to it. Thus it would be $$AbilityMod + ProfBonus*2$$
When Knowledge of Ages runs out
As soon as Knowledge of the Ages ran out, you completely lose proficiency in that skill. Since that skill no longer counts as one of your skill proficiencies you gain no benefit from Expertise (per the PHB). Thus, only Jack of All Trades applies and add half of your proficiency bonus to the ability check.
The calculation would now be $$AbilityMod + ProfBonus*0.5$$
Knowledge of Ages gets activated again
If you gain proficiency in that skill again (permanently or temporarily) then the skill is now a valid target for Expertise again. In this case, the calculation would go back to $$AbilityMod + ProfBonus*2$$