This question was asked before but closed. I can download the .dmg but Mac OS 10.8 does not recognize it.
*What’s the site you download from?
*What’s the size of your downloaded package?
*What’s the error message when you open it?
libreoffice.org, 276 kb, The following disk images couldn’t be opened: LibreOffice_4.03_MacOS_x86.dmg
oops, Reason, not recognized.
To provide an answer I will repeat what the OP posted in a comment about the details of the download:
276 kb, The following disk images couldn’t be opened: LibreOffice_4.03_MacOS_x86.dmg (emphasis added)
Thus the file has not downloaded properly.