Activation energy has a term that is proportional to temperature according to transition state theory (at least).

For instance, for the bimolecular reaction $\ce{A + B -> P}$, we have $E_\mathrm a = \Delta^{\ddagger}H^{\ominus} + nRT$ where $\Delta^{\ddagger}H^{\ominus}$ is standard enthalpy of activation, $n = 2$ for gas reaction and 1 for reaction in solution. This said, for many reactions, the activation enthalpy is much larger than $RT$ (which is ~$\pu{2.5 kJ/mol}$ at $\pu{300 K}$) and the activation energy can be approximately regarded as temperature independent.