Is it bad to declare a C-style string without const? If so, why?

Yes, this declaration is bad practice, because it allows many ways of accidentally provoking Undefined Behavior by writing to a string literal, including:

cool[0] = 'k';
strcpy(cool, "oops");

On the other hand, this is perfectly fine, since it allocates a non-const array of chars:

char cool[] = "cool";