  The Tsallis entropy, which is a generalization of the Boltzmann-Gibbs
entropy, plays a central role in nonextensive statistical mechanics of complex
systems. A lot of efforts have recently been made on establishing a dynamical
foundation for the Tsallis entropy. They are primarily concerned with nonlinear
dynamical systems at the edge of chaos. Here, it is shown by generalizing a
formulation of thermostatistics based on time averages recently proposed by
Carati [A. Carati, Physica A 348, 110 (2005)] that, whenever relevant, the
Tsallis entropy indexed by $q$ is temporally extensive: linear growth in time,
i.e., finite entropy production rate. Then, the universal bound on the entropy
production rate is shown to be $1/|1-q|$ . The property of the associated
probabilistic process, i.e., the sojourn time distribution, determining
randomness of motion in phase space is also analyzed.
