We consider membranes as fluid deformable surfaces and allow for higher order geometric terms in the bending energy related to the Gaussian curvature squared and the mean curvature minus the spontaneous curvature to the fourth power. The evolution equations are derived and numerically solved using surface finite elements. The two higher order geometric terms have different effects. While the Gaussian curvature squared term has a tendency to stabilize tubes and enhance the evolution towards equilibrium shapes, thereby facilitating rapid shape changes, the mean curvature minus the spontaneous curvature to the fourth power destabilizes tubes and leads to qualitatively different equilibrium shapes but also enhances the evolution. This is demonstrated in axisymmetric settings and fully three-dimensional simulations. We therefore postulate that not only surface viscosity but also higher order geometric terms in the bending energy contribute to rapid shape changes which are relevant for morphological changes of cells.