Consecutive pattern mining aiming at finding sequential patterns substrings, is a special case of frequent pattern mining and has been played a crucial role in many real world applications, especially in biological sequence analysis, time series analysis, and network log mining. Approximations, including insertions, deletions, and substitutions, between strings are widely used in biological sequence comparisons. However, most existing string pattern mining methods only consider hamming distance without insertions/deletions (indels). Little attention has been paid to the general approximate consecutive frequent pattern mining under edit distance, potentially due to the high computational complexity, particularly on DNA sequences with billions of base pairs. In this paper, we introduce an efficient solution to this problem. We first formulate the Maximal Approximate Consecutive Frequent Pattern Mining (MACFP) problem that identifies substring patterns under edit distance in a long query sequence. Then, we propose a novel algorithm with linear time complexity to check whether the support of a substring pattern is above a predefined threshold in the query sequence, thus greatly reducing the computational complexity of MACFP. With this fast decision algorithm, we can efficiently solve the original pattern discovery problem with several indexing and searching techniques. Comprehensive experiments on sequence pattern analysis and a study on cancer genomics application demonstrate the effectiveness and efficiency of our algorithm, compared to several existing methods.