In recent years the problem of orbital ordering and its effect on Mott-Hubbard metal-insulator transition has been attracted considerable interest. The problem comes to realize what role the orbital degree of freedom plays in driving the quantum phase transitions in strongly correlated systems. The various charge, spin and orbital degrees of freedom coexist or compete with each other. Since many different quantum phases are almost degenerate, a little change in orbital configuration usually leads to quantum phase transition. The minimal model including the orbital degree of freedom is the two-orbital degenerate Hubbard model. This model has been investigated with many methods and approaches (see for example [1-3]). This paper is to understand the orbital physics at finite temperature especially for the influences of the intraorbital and inter-orbital Coulomb interaction, orbital fluctuations and orbital ordering, the role of Hund's coupling in the phase transitions of the correlated systems with various electron fillings. We discuss the simple model with intra- and inter-orbital hopping of the electrons and investigate their influence on the renormalization. Our investigation is based on the diagrammatic theory for strongly correlated electron systems elaborated by us in previous time for orbital non degenerate systems [4-8]. As in the case of non degenerate systems the irreducible Green's functions for orbital degenerate systems have spin and frequencies conservation laws. The Dyson type equation has been established for one-particle renormalized Green's function and some simple approximations has been used to obtain the spectral function of the system.