  We study the finite temperature properties of a recently introduced string
bit model designed to capture some features of the emergent string in the
tensionless limit. The model consists of a pair of bosonic and fermionic bit
operators transforming in the adjoint representation of the color group SU(N).
Color confinement is not achieved as a dynamical effect, but instead is
enforced by an explicit singlet projection. At large N and finite temperature,
the model has a non trivial thermodynamics. In particular, there is a Hagedorn
type transition at a finite temperature $T=T_H$ where the string degrees of
freedom are liberated and the free energy gets a large contribution $\sim
N^{2}$ that plays the role of an order parameter. For $T>T_H$, the low
temperature phase becomes unstable. In the new phase, the thermodynamically
favoured configurations are characterized by a non-trivial gapped density of
the SU(N) angles associated with the singlet projection. We present an accurate
algorithm for the determination of the density profile at $N=\infty$. In
particular, we determine the gap endpoint at generic temperature and analytical
expansions valid near the Hagedorn transition as well as at high temperature.
The leading order corrections are characterized by non-trivial exponents that
are determined analytically and compared with explicit numerical calculations.
