  We investigate the relationship between shadow radius and microstructure for
a general static spherically symmetric black hole and confirm their close
connection. In this regard, we take the Reissner-Nordstr\"om (AdS) black hole
as an example to do the concrete analysis. On the other hand, we study for the
Kerr (AdS) black hole the relationship between its shadow and thermodynamics in
the aspects of phase transition and microstructure. Our results for the Kerr
(AdS) black hole show that the shadow radius $r_{\rm sh}$, the deformation
parameters $\delta_s$ and $k_s$, and the circularity deviation $\Delta C$ can
reflect the black hole thermodynamics. In addition, we give the constraints to
the relaxation time of the M$87^{*}$ black hole by combining its shadow data
and the Bekenstein-Hod universal bounds when the M$87^{*}$ is regarded as the
Reissner-Nordstr\"om or Kerr black hole. We predict that the minimum relaxation
times of M$87^{*}$ black hole and Sgr $A^{*}$ black hole are approximately 3
days and 2.64 minutes, respectively. Finally, we draw the first graph of the
minimum relaxation time $\tau_{\rm min}$ with respect to the maximum shadow
radius $ r_{\rm sh}^{\rm max}$ at different mass levels.
