Why does the equilibrium constant depend on the temperature but not on pressure and concentration?
Please have a look at this answer of mine. It contains the derivation of the defining formula of the equilibrium constant.

\begin{equation}
\log \underbrace{\prod_i [a_{i}]^{\nu_{i}}}_{= \, K} = -\frac{\Delta G^{0}}{RT} \qquad \Rightarrow \qquad \log K = -\frac{\Delta G^{0}}{RT} \ .
\end{equation}

This formula shows the temperature dependence explicitely, so it is clear that $K$ must depend on $T$.

Apart from the temperature this formula also relates $K$ to the standard Gibbs free energy of reaction $\Delta G^{0}$.
This quantity is defined for specific standard conditions involving standard pressure and equilibrium concentrations.
Since there aren't any other quantities contained in the formula (apart from the constant $R$) it follows that $K$ is defined for this specific pressure and these specific concentrations too and so it has no pressure or concentration dependence.