Is it true that there is no relationship between tempo and key?

It's not strictly true that there is no relationship between tempo and key, but in most traditional music, the correlation is not a fixed rule. In Western music, tempo and key are often chosen independently, and the relationship between them is more a matter of musical feeling than of any set formula.

*In some cases*, a faster tempo may suggest a more energetic, upbeat mood, while a slower tempo may imply a more relaxed or contemplative atmosphere. Similarly, certain keys may be associated with specific emotions or characteristics, such as the association of the key of A major with brightness and cheerfulness, or the key of C minor with melancholy or seriousness.

However, these associations are subjective and not always consistent, as different composers and performers may interpret and express the relationship between tempo and key differently. Ultimately, it's up to the composer, musician, or performer to decide how tempo and key interact to create the desired musical expression.