This is called "token authentication" and is supported by Devise, or can be relatively easily ginned up on your own. You want to generate a non-guessable secret token (your param), and then use that in lieu of a username. The devise wiki has links to a couple of examples:
https://github.com/plataformatec/devise/wiki/How-To:-Simple-Token-Authentication-Example
If you want a lighter-weight solution, you can also simply generate an auth token (using something like bcrypt) and then do something like:
@user = User.find_by_auth_token(params[:auth_token])
if @user is nil, then return a 403.