At what temperature do the laws of physics break down?
Hank Green is describing the concept of the Planck temperature, $$ T_\mathrm{P} = \sqrt{\frac{\hbar c^5}{Gk_B^2}}\approx1.4\times 10^{32}\:\mathrm K, $$ which is defined as $\frac{1}{k_B}$ times the Planck energy $E_\mathrm{P}=\sqrt{\hbar c^5/G}\approx 1.9\times 10^{9}\:\mathrm J$.
As with all the Planck units, we don't really know what happens at those scales, but we're pretty sure that the laws of physics as we know them are likely to require modifications to continue describing nature at some point before you reach that regime.
What doesn't happen at the Planck scale is that "the laws of physics break down", which is a meaningless catchphrase that shouldn't be used. Unless, in fact, the world changes so much that there is no regularity to physical phenomena and no way to predict how an experiment will pan out, even in principle, then what you have is not a breakdown of the laws of physics, it's just that you've left the region of validity of the laws you know, and you need to figure out what the laws are on the broader regime.
Adding to @Emilio's answer, what happens during Planck's temperature is unknown. Our laws of physics does not seem to work at that temperature(@EvilSnack) for e.g. gravitational force. At that temperature, gravitational force seems to become as strong as other fundamental forces like electromagnetic forces, strong and weak nuclear forces leading to research in quantum nature of gravitational forces. On the Wikipedia article of Planck's temperature:
At temperatures greater than or equal to $\mathrm{T_P}$, current physical theory breaks down because we lack a theory of quantum gravity.
This statement is explained in details in this site:
The Planck temperature is the highest temperature in conventional physics because conventional physics breaks down at that temperature. Above $\mathrm{10^{32}~K}$, Planck time-calculations show that strange things, unknown things, begin to happen to space and time. Theory predicts that particle energies become so large that the gravitational forces between them become as strong as any other forces. That is, gravity and the other three fundamental forces of the universe—electromagnetism and the strong and weak nuclear forces—become a single unified force. Knowing how that happens, the so-called "theory of everything," is the holy grail of theoretical physics today.
"We do not know enough about the quantum nature of gravitation even to speculate intelligently about the history of the universe before this time," writes Nobel laureate Steven Weinberg about this up-against-a-brick-wall instant in his book The First Three Minutes. "Thus, whatever other veils may have been lifted, there is one veil, at a temperature of $\mathrm{10^{32}~K}$, that still obscures our view of the earliest times." Until someone comes up with a widely accepted quantum theory of gravity, the Planck temperature, for conventional physicists like Steven Weinberg, will remain the highest temperature.
Basically, if $\mathrm{0~K}$ is absolute cold that Planck's temperature is absolute hot(i.e a body cannot get any hotter than Planck's temperature).