Current astronomical measurements indicate that approximately 73 \% of the universe is made up of dark energy .
Stochastically quantized self-interacting scalar fields can serve as suitable models to generate dark energy .
We study a particular model where the scalar field theory underlying dark energy exhibits strongest possible chaotic behaviour .
It is shown that the fluctuating chaotic field momenta obey a generalized ( nonextensive ) statistical mechanics , with an entropic index q given by q = 3 , respectively q = -1 if the escort formalism is used .