Question speaks for itself.
If you had an immortal (thinking God-Emperor Leto style), would language change? Assuming an immortal presence in society holding a high (or multiple high) positions of authority, I'd assume that language would not change as drastically as it could naturally.
Language naturally evolves, in the absence of forces that prevent it doing so. But there are cases where forces, such as national institutions that manage language, with strong government support, do indeed impede its evolution.
The classic example on Earth is Icelandic. Because of concerted efforts over the centuries, there is much less difference between the Icelandic of the middle ages and that of today, compared to most other European languages. (See https://en.wikipedia.org/wiki/Linguistic_purism_in_Icelandic )
Two other examples that come to mind are Hebrew and classical Latin, as practiced by their religions. The Latin that is used by the Catholic church today is the same as it was in the first few centuries AD, even as "street Latin" evolved into the Romance languages. Meanwhile, Hebrew completely died out as a spoken language, but was revived from its written texts.
What these examples have in common is a strong central authority invested in keeping the language from changing. If your immortal autocratc wants this to be the case, and invests in the structures to prevent language drift, I don't think it's far-fetched that language drift could be minimal over long periods of time.
Language would still change.
It's likely that the language of the mortals will over time diverge from the language of the god emperors. It will evolve into its own dialect and eventually its own language, because of all the reasons that languages in large societies evolve.
The language which the immortals speak will evolve as well. The speed of change will depend on the degree of isolation the immortals maintain (assuming there is a group, a community of immortals). There is ample evidence that the language spoken by small, isolated communities evolves much slower.
In this case the language of the immortals may take on the roles of Hebrew and Sanskrit in Judaism and Hinduism respectively: A language widely taught for religious or at least ritual purposes, which is the form of interaction with immortals; but not spoken in that form in everyday life, because it is too complicated, does not fit modern requirements, and because the language of the mortals simply evolved in the way languages evolve.
But still, the language of the immortals stays a spoken language, not a static text. Even the minds of immortals evolve. The circumstances will change as well. With them will change their language. I certainly speak differently than I did a decade ago. I notice how I pick up mannerisms and idiosyncrasies from people I meet: Using the same terms improves communication, they are cool, they apply well to a new situation etc. The languages of mortals and immortals will also co-evolve: The interaction between mortals and immortals is not a one-way street. But the language of the immortals will still evolve slower than the language of the mortals, not least because, as John remarked in a comment, young people are a driving force for change, also in language.
It all depends. As an author you can choose either way. A number of related questions:
New inventions and new phenomena will require new words. This is inevitable. But how much change is there in this society? That is up to the author. There is also possible to make old words have new meanings. ("to google" vs "to search the net")
Does the immortal themself want to appear to be one of the people. This will lead to them adopting the changes that occur naturally. If they want to appear above the people they will try to keep their language unchanged, and mostly succeed.
Is there TV or some other way for the immortal to speak to the people? And will they listen? This will contribute to keeping the popular language close to theirs.
Does people in general like and want to emulate the immortal? This will give stability. Or does people like the idea that the upper class doesn't understand them? This gives change, like cockney.
In general, how large is the differences between the classes? Large differences and strict separations lead to separate languages. The immortal will then only influence the upper class.
Religion can play a big part. If there is a strong church with the immortal either in charge or worshiped, or both, the clergy will speak like the boss does, unchanging throughout the ages. It is not a given that the populace will follow the clergy in this, but it could happen.
In short, make it up as you go along and nobody can tell you different. It is your world.
If the immortal is a true God (omniscient, perfected Being), there would be no reason to change the language other than for aesthetic variety, any language created by such a God already being perfect in all functional aspects.
Language would change the same. Your immortals at the top might still speak the ancient, classical form, and their administrators might understand it just as British Empire civil servants all had fluent Latin and Greek back in the day. But the people on the streets would still speak their fast-evolving urban slang incomprehensible to the old ones.
It would depend on what percentage of the population are immortal. If, say, half were immortal it would likely diminish the rate at which language would change. The phrasing of your question, though, leaves me with the impression that immortals would be an elite few. In this case I would expect the language to split into a conventional path, where the majority speak an evolving language, and a more isolated path, where the immortals use among themselves the language they learned as children with fewer changes from the original. (They would likely speak common as well, unless they chose to isolate themselves from mortals with translators.) After a few centuries, the more static version might become regarded as arcane or as a status symbol, and would at least be a sort of "secret code" they could use in public.
Captain Corelli's Mandolin, set in German occupied WWII Greece, has a hilarious character - not precisely immortal, but a British Secret Service agent.
Having been a public school boy, and a Classics scholar at Oxford or Cambridge, he drifted into his Secret Service role because of his ... fluency in Greek.
Unfortunately for him, Greek had changed in the last 2000 years, so, while (with an effort) the locals could just about understand him, and vice-versa, there was no way he could pass muster as a natural Greek speaker. This made him spectacularly useless as a spy. He had to remain as a practically mute shepherd living alone in the hills, while the locals pretended there was nothing out of the ordinary about this, to avoid blowing his cover in the face of the occupying army.
Consider this: do you use the same words you did years ago? (And I'm referring to more than just slang or colloquialisms.)
I mention this because over the last few decades I've noticed some phrases & words have fallen out of favor while others have come into common use. For example, "impact" has all but replaced "effect" (viz., "The news had a negative impact" instead of "The news had a negative effet"). And I'm amazed at how "Wait, what?" has come from nowhere to become ubiquitous. (I think the word people used to use in its place was "Huh?")
One factor in language evolution is that at various times languages change radically, while for long periods of time they change very slowly. As difficult as modern native English speakers may find reading Shakespeare or his contemporaries, if you look at literature written a few hundred years before that, English (or more properly Middle English) becomes a foreign language, with a different vocabulary & rules of grammar. Even Shakespeare's contemporaries had as much trouble understanding Middle English as we. (The poet Spencer misunderstood Middle English, & his attempt to write in Middle English comes across as at best an affected archaism, & at worst badly spelled Modern English!) The reasons for these sudden shifts is that the elite, who set the norms for usage & are conservative in changing these norms, isolated themselves from the mainstream, allowing the majority of speakers to radically change these norms. An example of this is the who/whom distinction, which has been dying for centuries. (Defoe, writing in the 18th century, frequently uses "who" as the object instead of "whom".) This would mean your immortals would speak an old-fashioned version of the language, although not as out-of-date as say Chaucer. maybe not even as out-of-date as Shakespeare or the King James Bible.
In short, language always changes, often too slowly for us to notice without paying attention. But with immortals around, interacting with the rest of us in daily life, there would be little opportunity for a sudden language shift.
