Canopus, the brightest and closest yellow supergiant to our Solar System, offers a unique laboratory for understanding the physics of evolved massive stars. The accurate and precise PIONIER data allowed us to simultaneously measure the angular diameter and the limb darkening (LD) profile using different analytical laws. We found that the power-law LD, being also in agreement with predictions from stellar atmosphere models, reproduces the interferometric data well. For this model we measured an angular diameter of $7.184 pm 0.0017 pm 0.029$ mas and an LD coefficient of $0.1438 pm 0.0015$, which are respectively $gtrsim 5$ and $sim15-25$ more precise than in our previous A&A paper on Canopus from 2008. From a dedicated analysis of the interferometric data, we also provide new constraints on the putative presence of weak surface inhomogeneities. Additionally, we analyzed the SED in a innovative way by simultaneously fitting the reddening-related parameters and the stellar effective temperature and gravity. We find that a model based on two effective temperatures is much better at reproducing the whole SED, from which we derived several parameters, including a new bolometric flux estimate. The Canopus angular diameter and LD measured in this work with PIONIER are the most precise to date, with a direct impact on several related fundamental parameters. Moreover, thanks to our joint analysis, we were able to determine a set of fundamental parameters that simultaneously reproduces both high-precision interferometric data and a good quality SED and, at the same time, agrees with stellar evolution models.
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