Choosing a quantity of interest
There is no quantity to check the sensitivity of in all models. What you should look at depends on the model and what it is used for. Here we outline three options: measures of model fit, predictions, and parameters.
Measures of model fit
If you are evaluating the model based on some measure, it can be useful to assess how this measure of performance changes when changing the prior or likelihood. Examples of measures of model fit include log-score, R2, and metrics such as MAE or RMSE.
Predictions
If you are interested in the predictions your model makes for some specific quantity, then you can look at how those predictions would change depending on the prior/likelihood perturbations.
Parameters
If your model has parameters that are meaningful and interpretable, then you can look at those parameters specifically. In many cases there are far too many parameters, or they are not interpretable individually, and the other options are likely more applicable.