In the last years, a lot of attention was attracted by the problem of page authority computation based on user browsing behavior. However, the proposed methods have a number of limitations. In particular, they run on a single snapshot of a user browsing graph ignoring substantially dynamic nature of user browsing activity, which makes such methods recency unaware. This paper proposes a new method for computing page importance, referred to as Fresh BrowseRank. The score of a page by our algorithm equals to the weight in a stationary distribution of a flexible random walk, which is controlled by recency-sensitive weights of vertices and edges. Our method generalizes some previous approaches, provides better capability for capturing the dynamics of the Web and users behavior, and overcomes essential limitations of BrowseRank. The experimental results demonstrate that our method enables to achieve more relevant and fresh ranking results than the classic BrowseRank.