Currently, when I start a search from the home page, I get to the Search Results page. That is fine - the search term is kept.
Then there is a link "Advanced Search". When I click on it, I get to the Advanced Search page (titled "Object Search"???) and the search term is replaced by "Enter search text here."

I consider this to be big usability flaw. I now know to copy my old search term in advance to the clipboard and then paste it again, but please "Don't make me think!".