Is there a way to find/search recently websites appeared in internet ? For example I want to find out latest games websites (not pages)?
No, not really.
There is no governing body where you have to report a new webpage or a new website. Thus there is no up-to-date central index of web sites nor a listing of new ones.
The closest thing we have are spiders who crawl the web. (Think Google, Ilse, Altavista, Coosto, ...). But those will only have partial listing and might not share that information.
Since you have a great answer from Hennes I'll explain why you may have to go down the new page route...
There is no way to class a new website in the manner you want IMO. For example, I can buy a new domain name, point it at my Windows Server and get the classic IIS7 screen... Well, this is a new website I would guess by definition but for this purpose it is not. However, the page responds, has links etc. And of course there is no limit to what the default website may be (IIS screen, an Apache message, custom message etc). I would suspect that a DNS change would also class as a new site.
So then assume I create the actual page 6 months later; well, this would not be picked up as the site isn't 'new'.
I think the page search is the only option... http://www.google.com/advanced_search
However, you'll still have an issue of getting results for just 1 category of website IMO (or at least, you'll have to accept a degree of inaccuracy).