We report experimental results for the temperature variance 2(z) and the corresponding frequency spectra P(f) in turbulent Rayleigh-Bénard convection (RBC) in a cylindrical sample of aspect ratioT= D/L = 1:00 (D = 1:12 m is the diameter and L = 1:12 m the height). The measurements were conducted in the Rayleigh-number range 1011 < Ra < 1:35 1014 and Pr ' 0:8. For Ra = 1:35x1014, 2(z) could be described well by a logarithmic dependence on the vertical position z in a range of z 1 < z < z 2 with z 1 ' 70 and z 2 = 0:1L. Here L=(2Nu) is the thickness of a thin thermal sublayer adjacent to the horizontal plate where the heat flux (denoted by the Nusselt number Nu) is carried mostly by thermal diffusion. In the log layer, we found that the temperature spectra had a significant frequency range over which P(f) f with close to 1. As Ra decreased, increased so that the log layer became thinner. At Ra = 2:05 1011, z 2 < z 1 and therefore there was no range for a log layer. Correspondingly, the temperature spectrum near the horizontal plate did not have the f1 scaling form either.