In this paper, we initiate the concern of enhancing the spoofing robustness of the automatic speaker verification (ASV) system, without the primary presence of a separate countermeasure module. We start from the standard ASV framework of the ASVspoof 2019 baseline and approach the problem from the back-end classifier based on probabilistic linear discriminant analysis. We employ three unsupervised domain adaptation techniques to optimize the back-end using the audio data in the training partition of the ASVspoof 2019 dataset. We demonstrate notable improvements on both logical and physical access scenarios, especially on the latter where the system is attacked by replayed audios, with a maximum of 36.1% and 5.3% relative improvement on bonafide and spoofed cases, respectively. We perform additional studies such as per-attack breakdown analysis, data composition, and integration with a countermeasure system at score-level with Gaussian back-end.