Here's a link to a dated paper describing the Sadoway lab approach 
(it's by Dane Boysen, a good researcher in Sadoway's lab, and is 
a more sober analysis giving more credit to prior work than is Sadoway's style).  
As with many things its a rebranding of an approach researched heavily previously.  
A major problem (I know first hand) is that running batteries hot accelerates corrosion in all kinds of ways.  
These liquid metal battery materials are also only actually cheap if they can be purchased at modest purity, but 
sometimes they need high purity to limit corrosion.  

So, a reasonable path of research but a real long shot, in my opinion.  
They've gotten nearly a decade out of the MIT hype machine, but I've not heard that they have the goods yet.

https://www.academia.edu/4779612/Liquid_Metal_Batteries_Past_Present_and_Future