Please read the cold junction temperature registers (0Ah and 0Bh) and check whether the linearized thermocouple temperature reading is fluctuating with the cold junction temperature. In the Electrical Characteristics Table of MAX31856, cold junction temperature error can be up to +/-0.7 degree when TA = -20 degree C to 85 degree C. As long as the cold-junction fluctuation is within this range from the expected cold-junction temperature at this temperature condition, this is within the datasheet specification.
To reduce the fluctuation, one option is to measure the cold-junction externally with a more precise temperature sensor and write the values to 8Ah(CJTH) and 8Bh(CJTL) registers.
The following GUI screenshot of MAX31856evsys is included as an example. The screenshot was taken when the evsys is placed on top of a foam on a desk, and left at ambient temperature. The thermocouple is immersed in a mug of water to serve as thermal mass. The sampling period is 100ms. Cold junction temperature(CJ Temp) is observed to be constantly fluctuating with value as low as 26.75 degree C and as large as 27.05 degree C.