Raised-Cosine Filter - phase response graphs for three kinds of filter
You need to realize that the term "raised cosine" says absolutely nothing about the related phase response. I think the context of this question of yours is audio equalization, and in that context filters with certain desired magnitude response shapes are sometimes described by piecewise constant values which are smoothly connected by raised cosine shapes.

So those shapes in your figures are idealized magnitude responses. Now you can use any filter design technique to approximate such a response. Still nothing is said about the phase, and it depends on the design technique which phase response you'll end up with. Analog filter designs will usually result in a minimum phase response. But note that this has absolutely nothing to do with the raised cosine shape of the desired magnitude response.

In digital communications raised cosine pulses are used because they satisfy the Nyquist criterion. There also the phase is important and the frequency response of the ideal pulse has a raised cosine magnitude shape and a linear phase.

In sum, it is important to understand that the "raised cosine" property is a description of the magnitude response, and the application and/or the filter design process will determine the phase response, which can be linear, minimum phase, or anything else.