The Galaxy Tab 3 is starting to see the Kit Kat update. The official OTA rollout has already started in France. The update has been captured and you can flash it to your device. Developer "Faria" has released a stock build that can be rooted. While the rom can be rooted it will install a locked bootloader. This means that you will not be able to flash back to previous versions. You will also lose the ability to flash custom recoveries and the only roms you will be able to run are stock touchwiz type roms. You can flash the rom without flashing the locked bootloader by flashing via mobile odin. The update brings all kinds of performance enhancements. You can root this rom via chainfires cf-autoroot. This rom also includes KNOX which adds extra security to your device and may trip if you try to root or moddify. You may want to wait for a recovery installable version. If you can't wait head to the link below to download the odin flashable rom!
Via Android.gs