What is the air temperature inside of a tire right after landing?
I'm not sure that there is a maximum design temperature and even if there was it wouldn't be that high, probably no more than 120C. 

The problem is not the temperature inside the tyre but actually the temperature internal to the carcass. The hot spots are around the bead area and around the cords on the shoulder of the tyre due to the loading. The rubber compounds will start degrading above 160C. This will be considered in the design of the tyre. 

The design will have been tested under representative speeds and loads on a rig and the temperatures checked to make sure that the temperatures are within the required limits. At the Goodyear Technical Centre they have a large steel flywheel which they rotate to represent landing speeds and then use a hydraulic ram to simulate the landing gear and push the tyre onto the flywheel. They cycle this several times and then measure the temperatures throughout the carcass. 

An under inflated tyre will suffer increased deflection of the carcass and be prone to over heating. Hence the main concern is then to maintain the correct operating pressure. The tyre will have charts specifying the correct pressure to use for the loading and landing speed.