Starting from a compact Finsler manifold, and using the Ricci scalar arising from the flag curvature tensor, the authors introduce the normalized Einstein-Hilbert functional on the analogy of the normalized Riemannain manifold. They derive the Euler-Lagrange equation for the Finslerian functional. In this equation, as a new phenomenon, the Landsberg curvature of the manifold also appears. The classical Riemann-Einstein metrics are critical points of the Finslerian functional, but non-Riemannian critical metrics are also presented.