Black holes emit radiation. They must, because they consume mass. Google "Hawking radiation."
The "zero temperature" systems in question are constrained somehow to limit their possible energy content to some finite maximum. The ratio dT/dE become negative. That's all it means. The laws of thermodynamics do not apply, because the conditions for applying them are not met.
The implied promise of free energy in the article is probably just the typical journalistic misinterpretation one finds in these science-for-dummies articles.
"Fully half the quotations found on the internet are either mis-attributed, or outright fabrications." - Abraham Lincoln