Using non-standard constants for SHA-256 For SHA-256, FIPS 180-4 states the following on the constants:


  These words represent the first thirty-two bits of the fractional parts of the cube roots of the first sixty-four prime numbers.


If one were to change some small number of those constants, say $n \leq 8$, to suitably "random" values, would this significantly weaken the cryptographic strength of SHA-256? The intent is simply to use a "non-standard" version of SHA-256.
Those are "nothing up my sleeve" numbers, their values aren't central to the security of the algorithm. Nevertheless an intent to arbitrarily choose different and thus incompatible values probably means something else you're doing is a bad idea.