ISO 14644 is one standard that can be used. There are equivalent standards for the EU, but which use Grade A, B, C, D, instead of ISO 5,6,7,8, etc.
Cold rooms are typically used for stability studies. I’m curious what your HEPA provided cold rooms are used for. The activity will help determine what ISO or Grade level you need to maintain. Assessing what the cleanliness level should be is the 1st step.