  The possible time variation of the fundamental constants of nature has been
an active subject of research in modern physics. In this paper, we propose a
new method to investigate such possible time variation of the speed of light
$c$ using the updated Hubble diagram of high-redshift standard candles
including Type Ia Supernovae (SNe Ia) and high-redshift quasars (based on UV-X
relation). Our findings show that the SNe Ia Pantheon sample, combined with
currently available sample of cosmic chronometers, would produce robust
constraints on the speed of light at the level of $c/c_0=1.03\pm0.03$. For the
Hubble diagram of UV+X ray quasars acting as a new type of standard candles, we
obtain $c/c_0=1.19\pm0.07$. Therefore, our results confirm that there is no
strong evidence for the deviation from the constant speed of light up to $z\sim
2$. Moreover, we discuss how our technique might be improved at much higher
redshifts ($z\sim5$), focusing on future measurements of the acceleration
parameter $X(z)$ with gravitational waves (GWs) from binary neutron star
mergers. In particular, in the framework of the second-generation space-based
GW detector, DECi-hertz Interferometer Gravitational-wave Observatory (DECIGO),
the speed of light is expected to be constrained with the precision of
$\Delta{c}/c=10^{-3}$.
