Android 6.0 marshmallow firmware for the Samsung Galaxy Tab 3 8.0 is available for download. This is an un official build and based on cyanogen mod 13. Keep in mind that not a whole lot works on this Galaxy Tab 3 8.0 Android 6 Marshmallow ROM. For instance, video camera does not work as expected and the developer has not tested calling and data features on it as T310. 
Steps for flashing the Galaxy Tab 3 8.0 Android 6 Marshmallow ROM:
– Make sure you’re running latest TWRP 2.8.7.0
– Copy GApps and ROM ZIP files to your internal or external SD Card
– Boot into Recovery
– Commit a factory reset / data wipe.
– Flash the ROM zip file
– Flash GApps zip
– Reboot
Flashing this will void warranty on your Samsung Galaxy Tab 3 8.0 so if your android tablet is under warranty, it is not suggested to flash this custom ROM.