  The charged vector $\rho$ mesons in the presence of external magnetic fields
at finite temperature $T$ and chemical potential $\mu$ have been investigated
in the framework of the Nambu--Jona-Lasinio model. We compute the masses of
charged $\rho$ mesons numerically as a function of the magnetic field for
different values of temperature and chemical potential. The self-energy of the
$\rho$ meson contains the quark-loop contribution, i.e. the leading order
contribution in $1/N_c$ expansion. The charged $\rho$ meson mass decreases with
the magnetic field and drops to zero at a critical magnetic field $eB_c$, which
means that the charged vector meson condensation, i.e. the electromagnetic
superconductor can be induced above the critical magnetic field. Surprisingly,
it is found that the charged $\rho$ condensation can even survive at high
temperature and density. At zero temperature, the critical magnetic field just
increases slightly with the chemical potential, which indicates that the
charged $\rho$ condensation might occur inside compact stars. At zero density,
in the temperature range $0.2-0.5~ {\rm GeV}$, the critical magnetic field for
charged $\rho$ condensation is in the range of $0.2-0.6~ {\rm GeV}^2$, which
indicates that the high temperature electromagnetic superconductor could be
created at LHC.
