Two (or more) URLs resulting in the display of the same content is what we call a "duplicate content problem."
Most Webmasters try to avoid that, as you are essentially competing against yourself for links and ranking.
If the new coolname.com domain is considerably better from a marketing viewpoint than your existing one, then switch the site to use the new domain -- It should recover from the temporary ranking loss after 30 days to nine months, depending on your current PageRank/link-popularity and how often your site is spidered.
Once you've done that, make sure that each of the pages on your site can be reached by one URL and one URL only; For example, if your site displays the same page at "mysite.com" and "www.mysite.com", then you've still got a problem. And you've got an even bigger problem if the same page is accessible at "anything-and-everything.mysite.com". You'll want to put some code on your server so that any request for a page in a non-canonical domain results in a 301-Moved Permanently redirect to the same page in your canonical domain. If your site is well-established, select the domain variation (www- vs. non-www) which has the most and best incoming links as your canonical domain.
Before fooling around with your domain, links, or any other 'technical factors' of your site, I strongly suggest you research the potential impact of doing so. Reading or asking here is a very good idea. Making your site appear on multiple domains is not.