A novel sparsely connected associative memory based on the preferential mechanism of heuristic annealed topology was proposed in this paper. Aimed at overcoming the disadvantage of quenched dilution as random synapses disconnection of the existing methods, this model, taking the ideology of annealed dilution of statistical physics into account, investigates the optimal synaptic dilution strategy under the constraints of limited metabolic energy, namely limited amount of neurons and connections. Based on explicit theoretical analysis, this model constructs a learning task-dependent network topology in a heuristic annealed way which is much closer to biological genuine system as possessing flexible adaptive topology. It can achieve better performance than the existing counterparts of the same class. The effectiveness and robustness of the proposed model is validated by a great number of experiments.