Hi Alex:
Welcome to the forum community.
My first suggestion would normally be
"check the radiator to ensure it is not obstructed by dirt or mud", but this is unlikely if your moto only has 2,500 kms on it. But check it anyway... spray some detergent directly into the radiator, then hose it out.
Next, check to ensure that the cooling fans operate by putting the bike on the center stand, starting the engine, and letting the motorcycle idle until the fans kick on. You should see the temperature gauge rise above the usual "3 bars", then the fans will start.
There is a rubber heat mat under the air cleaner - it is possible that it is misplaced, but again that is unlikely due to the low mileage.
What you should do is investigate inside the engine area to see if mice nests are creating blockages to air flow. A 10 year old motorcycle with only 2,500 kms has been sitting unused for many years... this makes the possibility of mice nesting in the engine area quite possible.
It is not difficult to disassemble all of the plastic body panels (we call them "Tupperware"), but it sure helps if you have an experienced person show you how to do it the first time. In the section of our forum titled
ST1300 Articles, you will find an article called
ST1300 - TupperWare Worksheet. Although this article does not contain disassembly instructions, it does show you what fastener goes where, which makes re-assembly quite easy.
Be aware that all of the fasteners on the motorcycle that appear to be Phillips head (cross-head) fasteners are not Phillips. They are a special type of fastener called a
JIS (Japanese Industrial Standard) screw. You need a JIS screwdriver to operate these fasteners, if you use a regular Phillips screwdriver on them you will chew the heads up. You can get a set of 3 of these for about €25, but you really only need the medium size. The most trusted manufacturer of JIS screwdrivers is Vessel - here is a link to their European website:
Vessel Europe.
A last possibility is that your moto is low on coolant. When you have the body panels off and when the engine is stone-cold, remove the radiator cap. The radiator should be full to the very top, and the coolant reservoir on the left side (just above the front of your left foot, behind the body panel) should be half-full.
Good luck.
Michael