How to Save FFT output for fast read-in?
That doesn't seem worth the bother. 

FFT of a 7 second long wave file on my windows 10 laptop using Matlab takes about 5 milliseconds. So unless you have a particularly slow setup, you will not save a significant amount of time by storing the FFTs instead of just reading the raw wave file and doing the processing again each time. 

That doesn't seem to justify the extra development effort for a non-standard file format.