This is actually a filesystem permission error and occurred from me on Mac OS X 10.10 (although I am system guy and probably screwed up my directory structure somewhere along the way).
I had to get into the nitty gritty. For me, the error was occurring on /var/folders (see your log in ~/Library/Logs/FlashPlayerInstallManager.log) . It reported an error with bogus file permissions for /var/folders/....
My solution was to
1) Login on command line as root.
2) backup the existing /var/folders directory (mv /var/folders /var/folders-bak)
3) mkdir /var/folders
4) chmod 777 /var/folders
5) Run install. Let it fail.
6) cd /var/folders
7) find . -exec chmod 777 {} \;
8) Re-run install. Install succeeded.
9) Restore /var/folders from previous (mv /var/folders-bak /var/folders)
Last step very important, or you'll likely be going to the Apple store to have someone boot up to terminal and restore original /var/folders (which even they will just likely want to restore your OS)
YMMV