Language is a result of brain function; thus, impairment in cognitive function can result in language disorders. Understanding the aging of brain functions in terms of language processing is crucial for modern aging societies. Previous studies have shown that language characteristics, such as verbal fluency, are associated with cognitive functions. However, the scaling laws in language in elderly people remain poorly understood. In the current study, we recorded large-scale data of one million words from group conversations among healthy elderly people and analyzed the relationship between spoken language and cognitive functions in terms of scaling laws, namely, Zipf's law and Heaps' law. We found that word patterns followed these scaling laws irrespective of cognitive function, and that the variations in Heaps' exponents were associated with cognitive function. Moreover, variations in Heaps' exponents were associated with the ratio of new words taken from the other participants' speech. These results indicate that the exponents of scaling laws in language are related to cognitive processes.