I am planning a level 10 dungeon crawl where I would like the final encounter to be an archmage. At the moment my party consists of a bard, a ranger, and a fighter with the knight subclass from UA (all level 10). I want this encounter to be difficult, if not extremely so, yet not impossible. According to XP calculators and such, this encounter is deadly however it is still within the "max xp" for the party.
Because I am not sure exactly how this will go and don't want either the party or the mage to be killed in the first couple rounds, I am having this encounter in a place where things could be added or subtracted from the fight rather easily. It is happening in a large throne room-type space and I figure that I should be able to feel out the battle after the first round and add other enemies if I need to. I also could simply choose not to use the wizard's high level spells in order to nerf the encounter on the fly.
Is the CR 12 archmage still too powerful for this party?
I have run this encounter. They will win easily.
I have run a very similar encounter, with 4 PCs at level 10 against a single Archmage, with all of her spells pre-cast on herself, as the first part of a multi-stage boss fight. As you describe, the battle with the archmage took place inside what was basically a large empty building. The players easily blew through this stage of the fight--even with Time Stop to set up defenses, the Archmage was soundly defeated without a single PC going down.
First, consider that while the Archmage has access to 9th level spells, all of her high-level spells are nondamaging. The strongest damage-dealing spell she has is only 5th level. While Time Stop is potentially useful, it's mostly good for letting her run away with Teleport. Additionally, she only has 99 HP, which means that if she does get caught in a focus fire from PCs, she's going down very quickly.
Second, the DMG has some guidelines for how difficult an encounter is, based on XP thresholds. An Archmage yields 8400 XP, and DMG 82 states that for 3 level 10 characters, the XP threshold for a "deadly" encounter is 8400 XP. That means the Archmage is barely on the edge of being "deadly," which in my experience usually means "one or two PCs actually take some damage". In fact, the daily adventure XP budget for 3 level 10 characters is 27,000 XP, or basically three Archmage encounters before a long rest.
Finally, a single-monster encounter is always on the losing side of the action economy, which further reduces the difficulty of the encounter. For example, if your bard manages to get a Hold Person off on the Archmage, the encounter is basically over, since the PCs get two consecutive turns against the Archmage.
How do you make it more difficult?
You should really be asking, "how do I make this encounter more difficult?". In my case, I made the archmage a multi-stage boss, as mentioned above. In the second stage, I gave her significantly improved survivability, by giving her a regenerating barrier with high HP, which forced the PCs to focus down a portion of the barrier before they could even touch the Archmage. Moreover, I gave her a number of minions under her control, to even out the action economy a bit more.
Overall, these changes did not make the encounter that much more deadly for the PCs, but it did force them to be more creative in their abilities and tactics than a straight fight in an empty room.