I know when I feel like Googling my t it is for a sense of connection.

Does the googling itself upset you, or is it the guilt and need/fear of telling your therapist about it?
The googling itself upsets me.
If the problem is not in the googling itself, or if it's not a big problem, I'd say google away, don't tell her, and try not to feel guilty.
That would never work for me. I LOOK guilty, LOL. She knew something was wrong when I couldn't look at her when I came to my session. I'm always honest with her.
I think, in a way, telling her that you're googling during a session ends up making that -portion- of the session about her. A couple of people have mentioned doing this as a way to maintain the connection. I don't see any problem with that. But I think in telling her about it, in a way, it makes it bigger than it is, and then the conversation ends up being about her, the information you found about her, how it makes you feel, etc. I know this goes against the adage of telling one's therapist everything, but just a thought about the potential gain in not telling her.
That's not the problem for me. She didn't even want to hear what information I found.
Of course if the core, underlying problem really is the googling itself, then I suppose you shouldn't do it. Though there's still the possibility of not telling her about it. Counterintuitive, I know, and I could be way off.

I also think it's an issue of seeking a connection, but not really getting one -- that's probably why it's making you so miserable. It's very frustrating but I think it's a good thing that you want to find a way to stop doing it, because doing it is making you unhappy.
Yes, it makes me unhappy and I am going to stop!!! I want to say that in the past I didn't google her very much at all because there wasn't anything about her, just her website and PsychologyToday, and things like that. I did google her H and kids, though. That's not for the connection, but it made me feel terrible too.
There are a couple things that make me feel connected to T when I'm not around him. I love to dance by myself when no one's around, so sometimes I'll put on a song that makes me think of him and dance to it. Or I'll doodle a little with him in mind, and maybe show the result to him later.
I need to paint more, because I can then bring in the picture and put in our "art gallery" in the office. That's just for the session; then I take it home again.
Another thing I started doing a while back, that I haven't done in a while now, is write about sessions afterward. This is when I was in really intense psychoanalysis, and it helped me think about and understand my sessions more deeply. Posting on this forum is my version of that.

Sometimes, when I needed to feel a connection, I would go back and read what I'd written about a particularly good session. There was one in particular I read over and over again when I needed to feel like things were ok between him and me. It helped me hang on to some of the more helpful things he said, and to remind me of the good feelings I had had. It was good to remind myself at times that that would always be there when I went back, even after really rough patches. And it helped I think in part because they were his words (as best I remembered them to transcribe later) and my words, and it was stuff about OUR connection. Not frustrating myself by trying to know more about him in ways that I am unlikely to ever actually know him. Feeling like the relationship we have now is good enough.

I do this too and it helps me a lot. I have a journal where I write things I find particularly helpful or comforting and I read over that between sessions sometimes. It's basically a big list of things he's said. He knows about it but doesn't know what's in it.