A problem is NP-hard iff every NP problem can be polynomially-time reduced to it.
Hardness is often intuitively explained as a lower bound. But it isn't, strictly speaking. For the sake of the argument, assume P=NP. Now the definition becomes:
A problem is P-hard iff every P problem can be polynomially-time reduced to it.
Because the above definition uses the polynomial-time reduction, the overall running time is polynomial (reduction + solving the resulting problem), no matter how easy is the resulting problem. Hence we could get an absurd result: a problem, which runs in constant time (hence lower bound is constant), is P-hard.
The definition makes total sense if we assume that NP>P. Do people assume NP>P?
The same question arises for the definition of PSPACE-hardness, where book authors use polynomial-time reduction, rather than something strictly easier than PSPACE.
I guess the answer to this question is simply "yes", sorry for the rant.