(2 J + 1) = 2 comes from the spin degeneracy. The summation over the Matsubara (odd-) frequencies is evaluated and the result reads, The first term is temperature independent. It leads to the divergence at zero temperature and can be renormalised in the standard way. Since, we are interested on the temperature dependent terms, the first term is trivially dropped. In the extremely hot medium, the system is supposed to take any flavor symmetry under the unitary representation. The mechanism is recognised as the flavor invariance in the extreme hot nuclear matter. Hence by the assumption that there is no specific flavor symmetry configuration is preferred, we do not need to project specific flavor symmetry. It is reasonable to believe that under extreme temperature such as the case of hadronic fireballs, the system is preferred to maintain the flavor chemical equilibrium rather than carries specific internal flavor symmetry. Under this assumption a specific flavor structure such as mesonic or baryonic Hagedorn states becomes unimportant and the classical Maxwell-Boltzmann statistics becomes satisfactory. Nonetheless, it is still possible to maintain the flavor and SU ( N$_{f}$ ) symmetry by consider- ing non-strangeness mesonic and baryonic Hagedorn states at low baryonic density. When the nuclear matter is compressed to higher baryonic chemical potential, the strange mesonic and baryonic Hagedorn states emerge in the system and so on. Furthermore, it is possible to imagine that under certain extreme hot and/or dense circumstances the color and flavor symmetries are modified to reproduce either the orthogonal or symplectic symmetry. The flavor degree of freedom is introduced trivially as follows 11 where ϵ$_{q}$ ( ⃗ p ) = √ ⃗ p 2 + m 2 $_{q}$. Only the real part is taken in Eq.(21). The resultant partition ln Z$_{q}$$_{¯}$ q = ℜ e   (2 J + 1) V N$_{f}$ ∑ q ∫ d $^{3}$⃗ p (2 π ) 3 ∑ i [ ln ( 1 + e − β ( ϵ$_{q}$ ( ⃗ p ) − µ$_{q}$ − i θ i $_{β}$) ) + ln ( 1 + e − β ( ϵ$_{q}$ ( ⃗ p )+ µ$_{q}$ + i θ i $_{β}$) )]) , (21) T ln Z$_{q}$$_{¯}$ q = 1 β ln Z$_{q}$$_{¯}$ $_{q}$, = ℜ e ( (2 J + 1) V ∫ d $^{3}$⃗ p (2 π ) 3 [ ϵ ( ⃗ p ) + 1 β ∑ i ln ( 1 + e − β ( ϵ ( ⃗ p ) − µ − i θ i $_{β}$) ) + 1 β ∑ i ln ( 1 + e − β ( ϵ ( ⃗ p )+ µ + i θ i $_{β}$) ) ]) . (20)