The linear thermal conductance of a quantum point contact displays a half-plateau structure, almost flat regions appearing close to half-integer multiples of the conductance quantum. This structure is investigated for the saddle-potential model; its behaviour as a function of contact parameters is also investigated. Half plateaus appear when the thermal energy is less than the subband separation and greater than the energy scale over which the transmission probability for a subband changes. The effect arises from the presence of a current node in the energy-resolved heat current, an energy at which the current is zero. When the transmission steps cross the current node as the gate voltage is varied, the heat conductance remains constant, creating the half plateaus, and this happens only for a certain temperature range. It is found that with increasing temperature the half plateaus become wider and flatter, which makes them more pronounced. It is also found that no half plateaus are present at the first step for any parameter values, and this is tied to the effect that the current node is pushed above the first step by the strong Seebeck potential.