Why doesn't a neutral solution at any temperature have pH 7? I am given 4 statements to choose from and find which is correct about a neutral solution: 


$K_\mathrm{w} = \pu{1.00\times10^-14 mol^2 dm^{-6}}$
The solution contains only $\ce{H2O}$
$\ce{[H+] = [OH-]}$
$\ce{pH =7}$


I can see that the first statement is wrong as that is only the case at $\pu{25 ^\circ C}$.  

The second statement is also incorrect - it can be when $\ce{H+}$ ions and $\ce{OH-}$ are equal in concentration. 
So I can see that option 3 is correct. 

However, I am confused why option 4 is wrong. 
Why is option 4 wrong?
As the temperature changes the value of $K_\mathrm{w}$ changes(as you wrote) and then the concentration of $\ce{H^+}$ and $\ce{OH^-}$ also changes and now the concentration of $\ce{H^+}$ will not remain $10^{-7}$ and hence the neutral point shifts. Therefore, at any temperature the $\ce{pH}$ of neutral point is not 7.