The paper presents a numerical solution to the problem of a hot rigid indenter sliding over a thermoelastic Winkler foundation with a thermal contact resistance at constant speed. It is shown analytically that no steady-state solution can exist for sufficiently high temperature or sufficiently small normal load or speed, regardless of the thermal contact resistance. However the steady state solution may exist in the same situation if the thermal contact resistance is considered. This means that the effect of the large values of temperature difference and small value of force or velocity which occur at no steady state can be lessened due to the thermal contact resistance. When there is no steady state, the predicted transient behavior involves regions of transient stationary contact interspersed with regions of separation regardless of the thermal contact resistance. Initially, the system typically exhibits a small number of relatively large contact and separation regions, but after the initial transient, the trailing edge of the contact area is only established and the leading edge loses contact, reducing the total extent of contact considerably. As time progresses, larger and larger numbers of small contact areas are established, unlit eventually the accuracy of the algorithm is limited by the discretization used.