We investigate analytically the effect of thermal conduction on the Kelvin-Helmholtz instability (KHI) in a straight pipe with different cross-sections. The results show that the relative tangential velocity of the interface between the upper and lower fluid in the pipe first increases and then decreases with the increase of the wave number. Furthermore, the smaller coefficient of interfacial heat conduction causes the relative tangential velocity to decrease considerably with the increase of the wave number, which is different from the behavior of the straight pipeline with the same cross-section. In addition, the heat conduction increases the growth rate of KHI, which is in accordance with the scenario of straight pipeline with the same cross-section.