  Generalized Noether's theory is a useful method for researching the modified
gravity theories about the conserved quantities and symmetries. A generally
Gauss-Bonnet gravity $f(R,\mathcal{G})$ theory was proposed as an alternative
gravity model. Through the generalized Noether symmetry, polynomial and product
forms of the $f(R,\mathcal{G})$ theory with corresponding conserved quantities
and symmetries are researched. Then suitable general forms of the polynomial
form $f(R,\mathcal{G}) \!=\! k_1 R^n + (6)^{\frac{n}{2}}(-1)^{n+1} k_1
\mathcal{G}^{\frac{n}{2}}$ and the product form $f(R,\mathcal{G}) \!=\! k ( R /
\sqrt{\mathcal{G}} )^n \mathcal{G}$ are found out, to contain the solution of
accelerated expansion cosmology. Both forms of $f(R,\mathcal{G})$ concerned in
this paper only possess time translational symmetry. And energy condition of
these solutions are also checked. To some extent, the consistency of
conservation of symmetry and energy condition is demonstrated. For the specific
form of different $n$, it needs further detailed study. Noting that, the
corresponding conserved quantities are both zero, and the only conservation
relation is conservation of energy.
