  The high temperature behaviour of the open bosonic string free energy in the
space $S^1 \otimes H^N$ with vanishingly small curvature is investigated. The
leading term of the high temperature expansion of the one-loop free energy,
near the Hagedorn instability, is obtained. The problem of infrared
regularization of thermodynamical quantities is pointed out. For minimally
coupling quantum fields related to the normal modes of strings, the results are
similar to the ones valid for Rindler space. In the lower mass string states
regime a connection with the quantum corrections to the black hole entropy is
outlined.
