  In the present paper we study the thermodynamical properties of finitely
generated continuous subgroup actions. We address a notion of topological
entropy and pressure functions that does not depend on the growth rate of the
semigroup and introduce strong and orbital specification properties, under
which, the semigroup actions have positive topological entropy and all points
are entropy points. Moreover, we study the convergence and Lipschitz regularity
of the pressure function and obtain relations between topological entropy and
exponential growth rate of periodic points in the context of semigroups of
expanding maps. The specification properties for semigroup actions and the
corresponding one for its generators and the action of push-forward maps is
also discussed.
