Algorithmic recourse seeks to provide actionable recommendations for individuals to overcome unfavorable outcomes made by automated decision-making systems. The individual then exerts time and effort to positively change their circumstances. Recourse recommendations should ideally be robust to reasonably small changes in the circumstances of the individual seeking recourse. In this work, we formulate the adversarially robust recourse problem and show that methods that offer minimally costly recourse fail to be robust. We restrict ourselves to linear classifiers, and show that the adversarially robust recourse problem reduces to the standard recourse problem for some modified classifier with a shifted decision boundary. Finally, we derive bounds on the extra cost incurred by individuals seeking robust recourse, and discuss how to regulate this cost between the individual and the decision-maker.