  The hadronic parameters of pseudoscalar ($\eta_c$) and scalar ($\chi_c$)
charmonium are determined at finite temperature from Hilbert moment QCD sum
rules. These parameters are the hadron mass, leptonic decay constant, total
width, and continuum threshold ($s_0$). Results for $s_0(T)$ in both channels
indicate that $s_0(T)$ starts approximately constant, and then it decreases
monotonically with increasing $T$ until it reaches the QCD threshold, $s_{th} =
4 m_Q^2$, at a critical temperature $T = T_c \simeq 180 \; \mbox{MeV}$
interpreted as the deconfinement temperature. The other hadronic parameters
behave qualitatively similarly to those of the $J/\psi$, as determined in this
same framework. The hadron mass is essentially constant, the total width is
initially independent of T, and after $T/T_c \simeq 0.80$ it begins to increase
with increasing $T$ up to $T/T_c \simeq 0.90 \; (0.95)$ for $\chi_c$
($\eta_c$), and subsequently it decreases sharply up to $T \simeq 0.94 \;
(0.99) \; T_c$, for $\chi_c$ ($\eta_c$), beyond which the sum rules are no
longer valid. The decay constant of $\chi_c$ at first remains basically flat up
to $T \simeq 0.80\; T_c$, then it starts to decrease up to $T \simeq 0.90
\;T_c$, and finally it increases sharply with increasing $T$. In the case of
$\eta_c$ the decay constant does not change up to $T \simeq 0.80 \;T_c$ where
it begins a gentle increase up to $T \simeq 0.95 \;T_c$ beyond which it
increases dramatically with increasing $T$. This behaviour contrasts with that
of light-light and heavy-light quark systems, and it suggests the survival of
the $\eta_c$ and the $\chi_c$ states beyond the critical temperature, as
already found for the $J/\psi$ from similar QCD sum rules. These conclusions
are very stable against changes in the critical temperature in the wide range
$T_c = 180 - 260 \; \mbox{MeV}$.
