score:7
If the liquid is red, it's mostly likely alcohol (ethanol), not mercury (so less of an issue- even if it broke it would be no worse than spilling a few drops of wine). Even small mercury thermometers are specifically allowed, as Johnny says:
By the TSA and some other authorities (but Cathay Pacific, for example, does not allow them).
So I think it should be okay, but you never know- you might lose it if some overzealous agent deems it a danger.