We derive a universal inequality on the unitary 2D CFT partition function with general central charge $c\geqslant 0$, using analytical modular bootstrap. We derive an iterative equation for the domain of validity of the bound on the mixed-temperature plane. The infinite iteration of this equation gives the boundary of maximal-validity domain of our inequality. In the $c\to\infty$ limit, with additional assumption of having a sparse spectrum below the scaling dimension $\frac{c}{12}+\varepsilon$ and below the twist $\frac{\alpha c}{12}$ (with $\alpha\in(0,1]$ fixed), our inequality implies that the grand-canonical free energy has universal large-c behavior in the maximal-validity domain, which does not encompass the entire mixed-temperature phase diagram, except in the case of $\alpha = 1$. In particular, we prove the conjecture proposed by Hartman, Keller and Stoica [1405.5137] (the $\alpha=1$ case): the free energy is universal in the large c limit for all $\beta_L\beta_R \neq 4\pi^2$.