  We present observations of oscillations in the shape of the temperature
profile of the large-scale circulation (LSC) of turbulent Rayleigh-B{\'e}nard
convection. Temperature measurements are broken down into Fourier moments as a
function of $\theta-\theta_0$, where $\theta$ is the azimuthal angle in a
horizontal plane at mid-height, and $\theta_0$ is the LSC orientation. The
oscillation structure is dominated by a 3rd order sine moment and 3rd order
cosine moment in a cubic cell. In contrast, these moments are not found to
oscillate in a cylindrical cell. This geometry-dependent behavior can be
explained by a model that assumes that the heat transported by the LSC is
conducted from the thermal boundary layers, and is proportional to pathlength
of the LSC along boundary layers at the top and bottom plates. In a
non-circular cross-section cell, oscillations of the LSC orientation $\theta_0$
result in an oscillation in the container shape in the reference frame of the
LSC, resulting in an oscillation in the pathlength of the LSC at a given
$\theta-\theta_0$. In a square-cross-section cell, this model predicts the
dominant 3rd order sine moment and 3rd order cosine moment with magnitudes
within 50\% of measured values, when using the amplitude of the oscillation of
$\theta_0$ as input. A cylindrical cell is special in that the pathlength is
independent of $\theta_0$, and so these oscillating moments are not induced. In
a cylindrical cell, the model reproduces the sinusoidal mean temperature
profile with a sloshing oscillation dominated by the 2nd order sine moment,
consistent with previous observations in that geometry.
