Gaussian Process Regression (GPR) is a widely used, non-parametric approach for prediction tasks in machine learning. It relies on positive definite kernels to model relationships between data points, but suffers from cubic computational complexity with respect to the number of data points. This issue arises during model fitting, where a dense linear system has to be solved, making GPR infeasible for large-scale datasets. A common approach to mitigate this issue is the use of low-rank approximations of the kernel matrix. Pivoted Cholesky factorizations provide an efficient mechanism for approximating symmetric positive definite matrices. Standard greedy and randomized pivot selection strategies are driven by residual diagonal entries and target the reduction of kernel approximation error. However, approximation quality does not always guarantee optimal predictive performance. In this talk, we introduce an objective-aware pivoting strategy derived for variational GPR and motivated by a statistical variational objective rather than a purely algebraic error measure. We derive a per-pivot gain criterion based on the variational free energy of a GPR model and show how it can be evaluated approximately using a Woodbury-based update while preserving the incremental structure of a pivoted Cholesky factorization. The resulting method yields low-rank factors tailored to the prediction task in GPR. Our numerical experiments show that variance-based pivoting can be suboptimal when large-variance directions are weakly relevant to the predictive objective, while objective-aware pivoting produces more effective low-rank approximations at small to moderate ranks. The results highlight the importance of aligning pivot selection criteria with problem-specific objectives and thereby demonstrating an objective-aware alternative to variance-based pivoting for predictive tasks.