Equation 7.vii in Nedelman and Jia, 1998 is used for this calculation:
var_sparse_auc(sparse_pk)A sparse_pk object from as_sparse_pk()
$$var\left(\hat{AUC}\right) = \sum\limits_{i=0}^m\left(\frac{w_i^2 s_i^2}{r_i}\right) + 2\sum\limits_{i<j}\left(\frac{w_i w_j r_{ij} s_{ij}}{r_i r_j}\right)$$
The degrees of freedom are calculated as described in equation 6 of the same paper.
Nedelman JR, Jia X. An extension of Satterthwaite’s approximation applied to pharmacokinetics. Journal of Biopharmaceutical Statistics. 1998;8(2):317-328. doi:10.1080/10543409808835241