In this study, we investigate the friction between a one-dimensional elastomer and a one-dimensional rigid randomly rough surface. Special emphasis is laid on the temperature dependence of the elastomer and its effect on the frictional behavior of the contact. The elastomer is modeled as a Kelvin body in a one-dimensional substitute model in the spirit of the method of dimensionality reduction. The randomly rough surface is a self-affine one-dimensional fractal. We provide a short discussion of a conical indenter pressed in a displacement controlled process into an elastomer. These analytical considerations are taken as a basis for the treatment of the randomly rough counter surface in contact to an elastomer with and without temperature dependent viscosity. We identify dimensionless quantities describing this process, introduce a thermal length scale, and give estimates for the coefficient of friction as function of velocity, indentation and thermal quantities.