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Equilibrium constants are ratios of the rate constants for the forward and reverse reactions, once the concentrations of reactants and products no longer change.
If you change the concentrations, the rate constants do not change once equilibrium has been re-established, because rate constants are what’s left of the observed rates after concentrations have been accounted for.
But changing the temperature can change rate constants, which are dependent on temperature; furthermore, it may (and often does) change the two different rate constants in different ways. (Since equilibrium constants are proportional to the logarithm of the temperature, you are guaranteed that the forward and reverse rate constants will be affected differently.)
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