Is there any regulation on the yearly renewal price of a domain under a premium TLD?
[A]m I just relying on industry/market self-regulation or otherwise?
Essentially, yes. Entities called Domain Registries (who are the people who oversee the day-to-day operations of Top Level Domains or "TLDs" such as .xyz) generally have most of the say over the pricing for any "premium" (newer) TLD they control.
In the case of .xyz specifically, a Registry called nic.xyz apparently owns the .xyz TLD and has control over its pricing. That Registry, in turn, is owned by a company called XYZ.
Registrars (assuming that they are a separate entity from the Registry itself) typically contract with Registries to sell a particular TLD or group of TLDs and add a small markup, as well as any "required fees". This price difference covers their own business of selling you the domain and any other responsibilities they might have.
Are yearly prices for premium TLDs universally regulated?
No, not in any meaningful way.
For "non-premium" (older) TLDs, per this Namecheap blog post:
Some domain names have fixed wholesale prices negotiated between the registry and a non-profit called the Internet Corporation for Assigned Names and Numbers (ICANN)[.] The wholesale price for older domain options like .com, .net and .biz are limited by [ICANN]. That’s why these domains are generally inexpensive to register.
However...
[ICANN] doesn’t control prices on any of the new TLD options that have appeared recently, such as .club, .shop, and .news. The wholesalers for these domain names can charge whatever they want for these new web addresses, making them subject to the open market rules of supply-and-demand and resulting in wide price ranges. [ICANN] also doesn’t limit what wholesalers of two-letter country domain options like .us and .ca can charge.
Renewals
Pricing for most common TLDs is pretty up front.
However, for newer TLDs such as .xyz, it is not uncommon to see a deceptively low introductory price and ridiculously high renewal rates (possibly thousands of dollars) simply because the Registries/ Registrars want to encourage adoption.
Simultaneously, for the Registries alone, there are the considerations that:
They typically only operate that TLD.
They are accounting for relatively low rates of likely registration.
Common generic TLDs (such as .com) are now saturated in terms of potentially available names (making newer TLDs more "valuable").
Secondary Domain Market
As pointed out in other comments and answers, there is also the issue of selling existing domains (e.g. coolname.tld) for a large price, independent of the associated TLD. These are sometimes referred to as premium domains and Registrars frequently facilitate these sales.
Often this is a one-time price (with any associated transfer fees) but some registrars will have "special renewal pricing" (whatever that might be) on these domains. These secondary sales are unregulated with the possible exception of any TLD/Registrar related requirements.
I think it is self regulated which is why you see registrars offer teaser rates to sign up and then the renewal is 10x what the original registration cost was. I would hesitate to put anything significant on a domain like that without knowing what the renewal terms would be like.
...yearly renewal price of a premium TLD?
I am looking at a.xyzdomain that is apparently considered 'premium'.
I think you may be confusing "premium TLD" with "premium domain". What you appear to be describing... "a .xyz domain advertised for $350 USD (for the first year?)" would seem to be a "premium domain", not a "premium TLD". (The other answers appear to have answered with respect to premium TLDs.)
The .xyz TLD simply isn't a "premium TLD". I used to have one. I initially acquired it for FREE for the first year (promo) but annual renewals were no more than a regular .com. A quick check now shows I can pick up a .xyz for $0.99 USD for the first year (another promo), but renewals are only around the $15 USD mark (which is even a little cheaper than .com domains for me). A ten year .xyz registration is about $130 USD.
However, a "premium domain" (mostly irrespective of TLD) is a domain that has already been registered and is currently for sale by a third party (possibly the registrar itself). The domain might have previously hosted a website or it might simply have a unique/desirable name. It is being marketed for it's potential "worth", which could be anything the market will stand. The $350 USD is resale value and transfer of ownership (which may or may not include an additional years registration). Once you have paid this initial fee and "bought" the domain then the renewal cost for additional years is the standard renewal cost for .xyz domains with that registrar*1 (which is probably why you don't see these costs listed here) - but you can transfer to another registrar at this stage if you wish.
*1 Unfortunately, domain registrars are often a bit unclear as to the costs of renewal. Personally, I would avoid any registrar that is not upfront about these costs.