Why is molarity independent of temperature?
First of all, molarity is temperature-dependent.

You might have meant molality, because molality is temperature-independent.

The reason is very simple, if you see molarity definition, molarity is the amount of substance of solute per litre of solvent and that mean it depends on volume.
You know that volume depends on temperature, therefore molarity is also temperature-dependent.  

Molality on the other hand is the amount of substance of the solute per mass of the solvent, both are independent of the temperature.