The entropic sampling Monte Carlo method within Wang-Landau algorithm is applied to investigate properties of a lattice model of strongly charged flexible 3-arm star-shaped polyelectrolyte. The density of states is calculated, from which the canonical properties of the system in a wide temperature range are obtained by simple integration. The effects of the arm length and the short-range monomer-monomer potential on the thermal and structural properties of star polyions are studied. We calculate such characteristics as mean square radius of gyration and its components, the radius vector of the center of mass, components of the tensor of inertia and parameters characterizing the shape of the polyion. In this work, we focus on how these characteristics are influenced by the change of the reduced temperature which, within the considered model, is a parameter combining the effect of real temperature, linear charge density and solvent dielectric permittivity. The coil-globule transition is observed in most of the considered cases, and for the polyions with the longest length of arms ( 24), the transition from a liquid globule to a solid-like state is observed. Comparison of polyelectrolyte models with neutral ones is given.