The "law of liberty" is seen twice in the New Testament, both times used by James in the first two chapters of his letter written to the twelve tribes of Israel which had been scattered to different regions. If doing an online search to find out what the law of liberty is ... like many other Bible topics ... be prepared to see no fewer than hundreds of different opinions. But as we springboard off the context from chapter 1 of his letter and move into chapter 2, we'll notice clearly how James is using this phrase as a reference to the law of Moses—something he also refers to as the royal law.
Within this context, he encourages his audience to follow all of that law and to speak "and do" as those who will be judged by that law—what he calls a law of liberty—but had just defined it as the Mosaic law. James would double-down to say faith without works is dead (useless) and that one cannot be justified by faith without works (of the law). He will declare that faith is perfected by works ... the precise opposite of what Paul said in his writing to the Galatians at the beginning of chapter 3. Even when using the example of Abraham, there is a conflict with what Paul said along the same lines, which we will cover more on the next program.