A natural property to demand from discrete in time approximations to gradient flows is energy stability: Just like the exact evolution, the approximate evolution should decrease the cost function from one time step to the next. Often, approximation schemes with desirable (e.g. unconditional) energy stability, such as minimizing movements, are only first order accurate in time. We will discuss general (problem independent) procedures for boosting the order of accuracy of existing implicit and semi-implicit variational schemes for gradient flows while preserving their desirable stability properties, such as unconditional energy stability. The resulting high order versions are formulated only in terms of multiple calls of the original scheme per time step, and therefore also essentially preserve their per time step complexity.