I went through the 1865-census for all of Norway, and there are very few candidates. The census was taken in the end of Dec 1865, so any child born that year would be in their 1st year.
Here is a candidate at the Kottum farm in Melhus, Sør-Trøndelag:http://digitalarkivet.uib.no/cgi-win/webcens.exe?slag=visbas...
That her patronym is Thomas would not be a surprise, taking the first son`s name into consideration: Albert Thomas.
The problem is that she is a fosterchild in the family (fosterchildren at the time were "always" related), so we don`t know who the parents are. But I am guessing/hoping that her father is Thomas Olsen, maybe son of the widow at Kottum, Berit Jensdtr., either that, or this Anne`s mother is one of Berit`s daughters. We need to check the churchregister for Melhus to find this out - I don`t have access to it, but I will try posting a message at the Norwegian Digital Archives, and see if anyone there does!