  We study thermodynamic of strange quark matter (SQM) using the analytic
expressions of free and internal energies. We investigate two regimes of the
high density and low density separately. As a vital program, in the case of a
massless gluon and massless quarks at finite temperature, we also present a
geometry of thermodynamics for the gluon and Bosons using a Legendre invariance
metric, it is so called as geometrothermodynamic (GTD) to better understanding
of the phase transition. The GTD metric and its second order scalar invariant
have been obtained, and we clarify the phase transition by study the
singularities of the scalar curvature of this Riemannian metric. This method is
ensemble dependence and to complete the phase transition. Meanwhile, we also
investigate enthalpy and entropy and internal energy representations. Our work
exposes new pictures of the nature of phase transitions in SQM.

General Relativity and Quantum Cosmology