  We investigate on the lattice the metastable confined phase above Tc in SU(N)
gauge theories, for N=8,10, and 12. In particular we focus on the decrease with
the temperature of the mass of the lightest state that couples to Polyakov
loops. We find that at T=Tc the corresponding effective string tension
\sigma_{eff}(T) is approximately half its value at T=0, and that as we increase
T beyond Tc, while remaining in the confined phase, \sigma_{eff}(T) continues
to decrease. We extrapolate \sigma_{eff}(T) to even higher temperatures, and
interpret the temperature where it vanishes as the Hagedorn temperature T_H.
For SU(12) we find that T_H/Tc=1.116(9), when we use the exponent of the
three-dimensional XY model for the extrapolation, which seems to be slightly
preferred over a mean-field exponent by our data.
