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) = 4m(Q)(2), at a critical temperature T = T-c similar or equal to 180 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 similar or equal to 0.80 it begins to increase with increasing T up to T/T-c similar or equal to 0.90(0.95) for chi(c) (eta(c)), and subsequently it decreases sharply up to T similar or equal to 0: 94(0: 99)T-c, for chi(c) (eta(c)), beyond which the sum rules are no longer valid.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 similar or equal to 0.80T(c), then it starts to decrease up to T similar or equal to 0.90T(c), and finally it increases sharply with increasing T. In the case of eta(c) the decay constant does not change up to T similar or equal to 0.80T(c) where it begins a gentle increase up to T similar or equal to 0.95T(c) beyond which it increases dramatically with increasing T. This behavior 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 MeV.