Update: GAME has responded with the following statement:
At GAME we guard our customers' details very carefully. We have thoroughly investigated the hacking claims made today by the website Pastebin, and can confirm that they are entirely false. The published email addresses are not registered users of GAME.co.uk, and there has been no breach of our database security.
We would like to assure all our customers that their details are well protected, and advise anyone who has any questions to contact our customer services team via the website, our Facebook page or Twitter account.
If you're a GAME.co.uk customer, you might want to change your password now.
According to a text file that's surfaced online, login details of accounts for the retailer's website have been obtained and posted online.
Of course there's no way to verify this without trying out some of the accounts, which isn't the most ethical thing to do, so just to be on the safe side you may want to change your password anyway.