The fundamental problem of thermoacoustic streaming on a cylinder in a strong standing acoustic field has been treated analytically. The cylinder is taken to be located at the velocity antinode of the plane standing wave, and also assumed to be compact (ok « 1). Only the high frequency limit is considered here for which the Stokes oscillatory boundary layer thickness is much smaller than the cylinder radius, and the dissipative effects in the boundary layer are more pronounced. It is found that the phased interaction of the first-order harmonic quantities in the boundary layer is capable of introducing a second-order time-averaged temperature distribution, in addition to the well known second-order time-averaged fluid motion. The associated steady temperature gradients cause localized heating and cooling effects over the surface of the cylinder, even in the complete absence of any externally applied temperature gradient. The role of little known second-order thermodynamic moduli is pointed out, which however do not contribute to this phenomenon for the case of an ideal gas host fluid. Results for this time-averaged thermal effect are presented and discussed with possible application to the acoustic heating of a suspension of small particles.