The Planck scale is the scale at which the effects of gravity are expected to become comparable to the other fundamental forces of nature, and it is defined by the Planck length, Planck time, and Planck energy. These values are incredibly small and represent the smallest possible length, time, and energy that can exist in the universe according to our current understanding of physics.
Interestingly, the speed of light squared is related to the Planck energy through the famous equation E=mc^2. This equation shows that energy (E) is equal to mass (m) times the speed of light squared (c^2). At the Planck energy scale, which is around 1.22 × 10^19 GeV (gigaelectronvolts), the energy required to create a black hole is reached.
Moreover, the Planck length can be derived from fundamental constants including the speed of light (c), the gravitational constant (G), and the reduced Planck constant (ħ), and is approximately equal to 1.616 x 10^-35 meters. Therefore, the speed of light squared also plays a role in the definition of the Planck length as it is one of the fundamental constants used to derive it.
In summary, the speed of light squared is related to the Planck scale through its connection to energy, as well as its role in the definition of the Planck length.