Skotty's Distributed Robocode Server
Linux doesn't keep stable ABI between kernel versions, so although it means binaries aren't compatible with old/new versions, it also means that a lot of cruft can be cleaned out, making it a more efficient, less bloated piece of software. Chances are, if you wanted you could take modern OpenJDK source and compile it against your old kernel. But there have been a lot of kernel fixes and improvements since F13, so it's probably worth the upgrade from a security standpoint anyway for anything that will be public-facing.
Actually, that's not really accurate. The Linux Kernel keeps the ABI to userspace *very* strictly stable. The part of the kernel that doesn't have a stable ABI is the interface to drivers, but that's not relevant to the issue of programs like Java 7. In this context, the issue is that certain userspace libraries that don't always maintain a stable ABI.
The thing with linux distributions, is that they don't just contain the core OS, they also contain libraries that aren't always compatible across versions. On Windows, similar libraries often are not compatible across versions either (i.e. MSVC redistributables which are essentially libc shared libraries). The only thing that's different under Linux distributions, is that under Linux, the distribution is typically responsible for providing those libraries that may not be compatible across versions, whereas under Windows individual application installers are responsible for bringing in their own versions of every library. The advantage is that it makes is that the system is not filled with multiple versions of the same library and bugfixes in a library that many programs depend on can be taken advantage of much more rapidly, among other advantages. The disadvantage is that you cannot treat major versions of Linux distributions as compatible unless you bundle your libraries the way Windows programs typically do (which you can, and is a common practice if one wants a program to work across versions). In summary: It's not a deficiency of linux distributions, it's just a different philosophy in how programs are bundled, with both advantages and disadvantages. One thing this means, is that you shouldn't expect to be able to be able to use the latest versions of software unless you either keep your distribution version up-to-date, or are willing to compile the software yourself.