Principal Component Analysis as a step between STFT and MFCC For computing Mel Frequency Cepstral Coefficients you can use already calculated STFTs as a basis and perform the Mel frequency mapping on it.

My question: Does it make a significant difference if I calculate the STFTs, perform a PCA-transform on them and then calculate the MFCC compared to computing the PCA-transform at the very end on the MFCCs?

In both cases we have two dimensionality reductions (PCA and MFCC) and I'm not sure weather the order makes a difference.
One dimensionality reduction method (PCA) is linear, the other (MFCC) is  not. It  is thus unlikely that there is no difference at all since they probably do not commute. One serves more decorrelation, the other feature extraction. 

The importance of the order thus depends a lot on your signal class, and what your purpose is.