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";

Yes, in C++ you should always refer to string literals with variables of type const char * or const char [N]. This is also best practice when writing new C code.

String literals are stored in read-only memory, when this is possible; their type is properly const-qualified. C, but not C++, includes a backward compatibility wart where the compiler gives them the type char [N] even though they are stored in read-only memory. This is because string literals are older than the const qualifier. const was invented in the run-up to what's now called "C89" -- the earlier "K&R" form of the language did not have it.

Some C compilers include an optional mode in which the backward compatibility wart is disabled, and char *foo = "..."; will get you the same or a similar diagnostic that it does in C++. GCC spells this mode -Wwrite-strings. I highly recommend it for new code; however, turning it on for old code is liable to require an enormous amount of scutwork for very little benefit.

It's bad. It's very bad. To the point this isn't possible to do anymore in C++11.

Modifying the memory of a string literal is undefined behaviour.