= Solution
Count oscillator polarizations at fixed center-of-mass <momentum>. In a critical <closed bosonic string>, each chiral sector has 24 transverse oscillator species. A species at frequency $m$ can be occupied $0,1,2,\ldots$ times, so its contribution is $(1-q^m)^{-1}$. The full chiral <generating function> is therefore
$$
F(q)=\sum_{N\ge0}d_Nq^N=\prod_{m\ge1}(1-q^m)^{-24}.
$$
The two sectors are independent, while <closed-string level matching> restricts their oscillator numbers to the same $N$. Thus the number of states at the closed-string mass level is $g_N=d_N^2$, not $d_{2N}$.
Put $q=e^{-\beta}$, with $\beta>0$. The <Dedekind eta function> obeys
$$
\eta\left(\frac{i\beta}{2\pi}\right)=e^{-\beta/24}\prod_{m\ge1}(1-e^{-\beta m}),\qquad \eta(-1/\tau)=\sqrt{-i\tau}\,\eta(\tau).
$$
The transformed argument $2\pi i/\beta$ has large <imaginary part>, so its product approaches one and $\eta(2\pi i/\beta)\sim e^{-\pi^2/(6\beta)}$. Hence
$$
F(e^{-\beta})\sim\left(\frac{\beta}{2\pi}\right)^{12}\exp\left(\frac{4\pi^2}{\beta}-\beta\right).
$$
This establishes the singular exponential as well as its power prefactor. The corrections to the transformed eta product are exponentially small in $1/\beta$.
The <Cauchy coefficient formula>, followed by $q=e^{-\beta}$, gives a steepest-descent integral with leading form
$$
d_N\sim\frac1{2\pi i}\int d\beta\,\left(\frac\beta{2\pi}\right)^{12}\exp\left((N-1)\beta+\frac{4\pi^2}{\beta}\right).
$$
Let $n=N-1$. The saddle of $S(\beta)=n\beta+4\pi^2/\beta$ is
$$
\beta_* =\frac{2\pi}{\sqrt n},\qquad S(\beta_*)=4\pi\sqrt n,\qquad S''(\beta_*)=\frac{n^{3/2}}\pi.
$$
Along the vertical steepest-descent direction $\beta=\beta_*+it$, the quadratic part is $-S''(\beta_*)t^2/2$. Thus the <saddle-point approximation> gives the Gaussian prefactor
$$
\frac{(\beta_*/2\pi)^{12}}{\sqrt{2\pi S''(\beta_*)}}=\frac1{\sqrt2}n^{-27/4}.
$$
The point $q=1$ supplies the largest exponential contribution. At another fixed <root of unity> of order $k>1$, the corresponding modular singularity gives a smaller exponential $e^{4\pi\sqrt n/k}$, so it does not change this leading asymptotic. Squaring the chiral result gives the <large-level degeneracy of a closed bosonic string>
$$
\boxed{d_N\sim\frac1{\sqrt2}(N-1)^{-27/4}e^{4\pi\sqrt{N-1}},\qquad g_N\sim\frac12(N-1)^{-27/2}e^{8\pi\sqrt{N-1}}.}
$$
Replacing $N-1$ by $N$ gives the equivalent usual leading asymptotic. The power $-27/2$ matters: the two-sector degeneracy is not just an unspecified exponential.
Since $M=2\sqrt{(N-1)/\alpha'}$, the exponential is $e^{\beta_H M}$ with
$$
\boxed{\beta_H=4\pi\sqrt{\alpha'},\qquad T_H=\frac1{4\pi\sqrt{\alpha'}}.}
$$
Here Boltzmann's constant is one. Coarse-graining the levels into a density per unit <rest mass> introduces the Jacobian $dN/dM=\alpha'M/2$. Consequently
$$
\rho(M)\sim C M^{-26}e^{\beta_HM},\qquad C=2^{25}\alpha'^{-25/2}.
$$
This is distinct from the degeneracy per level, whose power in $M$ is $-27$. The internal-state contribution to a <canonical partition function> behaves at large mass as
$$
Z_{\mathrm{internal}}(\beta)\sim\int^\infty dM\,C M^{-26}e^{-(\beta-\beta_H)M}.
$$
It converges for $\beta>\beta_H$ and diverges for $\beta<\beta_H$. At $\beta=\beta_H$, this particular internal rest-mass integral converges because of the power $M^{-26}$; one should not discard the prefactor and claim divergence there without specifying the full thermodynamic ensemble. For example, including continuous <momenta> in 25 spatial dimensions multiplies the high-mass integrand by a factor proportional to $M^{25/2}$, still leaving convergence of this ideal one-string integral at the endpoint, although sufficiently high <derivatives> are singular. Multi-string effects, volume and interactions affect the detailed limiting thermodynamics.
The leading microcanonical entropy grows as $S(E)\sim\beta_HE$, with logarithmic corrections, so increasingly large energies are stored in increasingly excited, long strings rather than in an arbitrarily hot ordinary gas. The <Hagedorn temperature> is the boundary of canonical convergence from above in <inverse temperature>; extrapolating the free-string canonical description to $T>T_H$ fails. This counting does not settle the interacting phase above that scale. Also, the bosonic ground-state <tachyon> is a separate vacuum instability; the high-level asymptotic is meaningful as a formal spectrum calculation without asserting a stable bosonic thermal vacuum.
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