The maximum traveling salesman problem (Max TSP) consists of finding a Hamiltonian cycle with the maximum total weight of the edges in a given complete weighted graph. This problem is APX-hard in the general metric case but admits polynomial-time approximation schemes in the geometric setting, when the edge weights are induced by a vector norm in fixed-dimensional real space. We propose the first approximation scheme for Max TSP in an arbitrary metric space of fixed doubling dimension. The proposed algorithm implements an efficient PTAS which, for any fixed $\varepsilon\in(0,1)$, computes a $(1-\varepsilon)$-approximate solution of the problem in cubic time. Additionally, we suggest a cubic-time algorithm which finds asymptotically optimal solutions of the metric Max TSP in fixed and sublogarithmic doubling dimensions.