Why Does Activation Energy not change with Temperature change
You have conceptual misunderstanding. In the activation energy context, energy of reactants, products and transition state are all potential energies, independent on the kinetic energy of molecules.
The kinetic energy of molecules is used to reach the peak of the potential energy, similarly as e.g. a acrobatic aircraft uses its kinetic energy to climb to the apex of an acrobatic loop.
Higher temperature means higher kinetic energy, but it does not mean the potential energy barrier is lower. It means there is higher probability to overcome it, leading to a faster reaction.
There is a famour Arrhenius' equation for the reaction rate:
$$k = A \exp{\left(\frac{-E_\mathrm{A}}{kT}\right)}$$
where $A$ is the frequency factor, describing the rate of molecular collisions, potentially able to take part of the reaction and
$\exp{\left(\frac{-E_\mathrm{A}}{kT}\right)}$ is the term from the Boltzmann distribution,
telling us the probability a molecule at temperature $T$ has sufficient energy, if the activation energy is $E_\mathrm{A}$, and $k$ is the Boltzmann constant.