How Does the RMS of White Noise Change with Sampling Frequency?
It changes according to the bandwidth of the LPF Filter in the sampling chain.
Namely if you use the same LPF yet with two different samplings rates the RMS of the noise is the same.

Now, the theoretical RMS of White Noise is given by its PSD which is a constant.
The RMS of the sampled noise is the constant value multiplied by the BW of the LPF in the sampling chain.