I can easily understand both sides of this in spite of my own opinion. If it disturbs you, maybe do the usual thing and discuss it with the T why, tell her your view on its being unethical etc. Might be a good topic and perhaps the T would learn something from it as well.

The truth though is that there is no way to enforce this type of ethic unless it is related to some extremely serious criminal act (e.g. the T would use the info to breach confidentiality or in other ways) - everyone will google privately whatever they want these days, regardless of what others say, I believe. You can tell the T it bothers you and maybe they will never do it again or perhaps they will just be more careful not to make it known.