We address the context of Single-Audience Value-Based Abstract Argumentation Framework (AVAF), where the arguments are labeled with the social values that they promote and the activation/deactivation of the attacks depends on the audience profile (expressed as a set of preferences between the social values). Herein, we introduce a new notion of robustness for measuring the sensitivity of the outcome of the reasoning to the extent of changes in the audience profile. In particular, for a set of arguments S or a single argument a, we define the robustness degree of the status of S or a as the maximum number k* of deletions/insertions of preferences from/into the audience profile that are tolerable, in the sense that S remains an extension (or a non-extension) or a accepted (or unaccepted) after performing at most k* deletions/insertions. We introduce the decision problems related to the computation of the robustness degree and focus on thoroughly investigating their computational complexity.