Summary: The purpose of this paper is to determine the large \(N\) asymptotics of the free energy \(F_N(a,U| A)\) of \(YM_2\) (two-dimensional Yang Mills theory) with gauge group \(G_N= \text{SU}(N)\) on a cylinder where \(a\) is a so-called principal element of type \(\rho\). Mathematically, \[ F_N(U_1,U_2| A)= \frac{1}{N^2} \log H_{G_N} (A/2N,U_1,U_2), \] where \(H_{G_N}\) is the central heat kernel of \(G_N\). We find that \[ F_N(a_N,U_N| A)\sim \frac NA \Xi(d\theta, d\sigma), \] where \(\Xi\) is an explicit quadratic functional in the limit distribution \(d\Sigma\) of eigenvalues of \(U_N\), depending only on the integral geometry of SU(2). The factor of \(N\) contradicts some predictions in the physics literature on the large \(N\) limit of \(YM_2\) on the cylinder (due to Gross-Matytsin, Kazakov-Wynter, and others).