Can we know about black hole thermodynamics through shadows?

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 $r_{\mathrm{s}\mathrm{h}}$, the deformation parameters ${\delta }_s$ and $k_s$, and the circularity deviation $\mathrm{\Delta }C$ can reflect the black hole thermodynamics. In addition, we give the constraints to the relaxation time of the M${87}^{\ast }$ black hole by combining its shadow data and the Bekenstein-Hod universal bounds when the M${87}^{\ast }$ is regarded as the Reissner-Nordstr"om or Kerr black hole. We predict that the minimum relaxation times of M${87}^{\ast }$ black hole and Sgr $A^{\ast }$ black hole are approximately 3 days and 2.64 minutes, respectively. Finally, we draw the first graph of the minimum relaxation time ${\tau }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ with respect to the maximum shadow radius $r_{\mathrm{s}\mathrm{h}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ at different mass levels.