Activation energy has a term that is proportional to temperature according to transition state theory (at least). This relies on the definition of activation energy as the parameter

$$E_\mathrm a = RT^2\left(\frac{\mathrm d \ln k}{\mathrm dT}\right)$$

(see [this answer](https://chemistry.stackexchange.com/a/15419/16683)).

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, and $n = 2$ for a gas-phase reaction and $1$ for a reaction in solution. That 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.