I never heard that talking point when Rand Paul and others filibustered on drones and the NSA.
I think the method in the madness is to make it a more visible issue so they can capitalize on it later. Otherwise I don't think 29 senators would be doing it. More like 5. We'll just have to see though.
There is nothing for the Democrats to capitalize. In another decade or two, temperatures will probably be the same as they are now (same for the last 10,000 years), and the hoax will be revealed. The weirdest thing about this era is the lack of climate change.
The change itself probably won't be particularly noticeable. It doesn't take a big change in temperature to mess up the global ecosystem.
