  We investigate the fluctuations of the free energy of the $2$-spin spherical
Sherrington-Kirkpatrick model at critical temperature $\beta_c = 1$. When
$\beta = 1$ we find asymptotic Gaussian fluctuations with variance
$\frac{1}{6N^2} \log(N)$, confirming in the spherical case a physics prediction
for the SK model with Ising spins. We furthermore prove the existence of a
critical window on the scale $\beta = 1 +\alpha \sqrt{ \log(N) } N^{-1/3}$. For
any $\alpha \in \mathbb{R}$ we show that the fluctuations are at most order
$\sqrt{ \log(N) } / N$, in the sense of tightness. If $ \alpha \to \infty$ at
any rate as $N \to \infty$ then, properly normalized, the fluctuations converge
to the Tracy-Widom$_1$ distribution. If $ \alpha \to 0$ at any rate as $N \to
\infty$ or $ \alpha <0$ is fixed, the fluctuations are asymptotically Gaussian
as in the $\alpha=0$ case. In determining the fluctuations, we apply a recent
result of Lambert and Paquette on the behavior of the Gaussian-$\beta$-ensemble
at the spectral edge.
