  The squared momentum transfer spectra of $\eta$ and $\eta ^0$, produced in
high-energy photon-proton ($\gamma p$) $\rightarrow \eta(\eta^0)+p$ processes
in electron-proton ($ep$) collisions performed at CEBAF, NINA, CEA, SLAC, DESY,
and WLS are analyzed. The Monte Carlo calculations are used in the analysis of
the squared momentum transfer spectra, where the transfer undergoes from the
incident $\gamma$ to emitted $\eta(\eta^0)$ or equivalently from the target
proton to emitted proton. In the calculations, the Erlang distribution and
Tsallis-Levy function are used to describe the transverse momentum ($p_T$)
spectra of emitted particles. Our results show that the average transverse
momentum ($\langle p_T\rangle$), the initial-state temperature ($T_i$), and the
final-state temperature ($T_0$) roughly decrease from the lower center-of-mass
energy ($W$) to the higher one in the concerned energy range of a few GeV,
which is different from the excitation function from heavy-ion collisions in
the similar energy range.
