This work addresses the problem of temporal action localization with Variance-Aware Networks (VAN), i.e., DNNs that use second-order statistics in the input and/or the output of regression tasks. We first propose a network (VANp) that when presented with the second-order statistics of the input, i.e., each sample has a mean and a variance, it propagates the mean and the variance throughout the network to deliver outputs with second order statistics. In this framework, both the input and the output could be interpreted as Gaussians. To do so, we derive differentiable analytic solutions, or reasonable approximations, to propagate across commonly used NN layers. To train the network, we define a differentiable loss based on the KL-divergence between the predicted Gaussian and a Gaussian around the ground truth action borders, and use standard back-propagation. Importantly, the variances propagation in VANp does not require any additional parameters, and during testing, does not require any additional computations either. In action localization, the means and the variances of the input are computed at pooling operations, that are typically used to bring arbitrarily long videos to a vector with fixed dimensions. Second, we propose two alternative formulations that augment the first (respectively, the last) layer of a regression network with additional parameters so as to take in the input (respectively, predict in the output) both means and variances. Results in the action localization problem show that the incorporation of second order statistics improves over the baseline network, and that VANp surpasses the accuracy of virtually all other two-stage networks without involving any additional parameters.