  We study the spectra of two-gluon glueballs and three-gluon oddballs and
corresponding equation of state in $5$-dimensional deformed holographic QCD
models in the graviton-dilaton system, where the metric, the dilaton field and
dilaton potential are self-consistently solved from each other through the
Einstein field equations and the equation of motion of the dilaton field. We
compare the models by inputting the dilaton field, inputting the deformed
metric and inputting the dilaton potential, and find that with only 2
parameters, the $5$-dimensional holographic QCD model predictions on
glueballs/oddballs spectra in general are in good agreement with lattice
results except two oddballs $0^{+-}$ and $2^{+-}$. From the results of
glueballs/oddballs spectra at zero temperature and the equation of state at
finite temperature, we observe that the model with quadratic dilaton field can
simultaneously describe glueballs/oddballs spectra as well as equation of state
of pure gluon system. The model with quadratic $A_{E}(z)$ can describe
glueballs/oddballs spectra, but its corresponding equation of state behaves
more like $N_{f}=2+1$ quark matter. These are consistent with dimension
analysis at UV boundary.
