Barrier height is not equal with the activation energy. In some case there are no potential barrier, but the reaction has activation energy (in some case the activation energy is negative). According to the definition:

Ea= -∂ln(k)/ ∂(1/RT)

Activation energy is the local slope of the Arrhenius plot of an elementary step. The Arrhenius plot is not always a straight line. It includes all dynamical effects of the molecular processes (not only the energetics, there are the internal and external motion of molecules (rotation, translation, vibration)). Activation energy is not a simple additive quantity.

You can measure something in a gas mixture...but activation energy makes sense only for elementary reactions, not for a whole mechanism.

What do you mean by this:

"What if the reaction rate goes down"? What is the variable when reaction rate will decrease (temperature, reactants, pressure)?

@ Peter Szabo, I mean the process in which the reaction rate decreases when surfactant is added to the mixture. I suppose that in such a case the activation energy should increase (or does not change) under influence of surfactant, but actually it decreases.

@ Gert Van der Zwan, thank you for the references.

Yes, activation energy is am experimentally determined index of the temperature dependence of a reaction. It is tempting to identify Ea with an elementary reaction (such as the rate determining step) but the energy requirements for a successful collision depend on the type as well as total energy. Nevertheless, the Arrhenius equation does suggest some exponential dependence of rate constant with T.

Activation energy makes sense here for each stage separately. Exponential functions with different bases and/or exponents cannot be added together.