t-butylbenzene in highly acidic medium Why would the following mechanism be valid for the conversion of t-butylbenzene to benzene in highly acidic medium followed by an aqueous workup? 

The main problem I have with this mechanism is the formation of the secondary doubly allylic carbocation. Why not go for the tertiary, inductively and doubly allylic stabilized carbocation? 

But I guess the problem with going down this path it becomes hard to see how the t-butyl group might leave. 



As you can see, the book's mechanism is entirely dependent on having the secondary allylic carbocation.
All 6 positions on the benzene ring are being protonated (the frequency of protonation at the different sites is related to the stability of the carbocation generated) in acidic media, so all of the ions you suggested are being formed.  Protonation at 5 of these positions can only regenerate starting material when one of the two protons at the initial site of protonation is removed (an identity reaction, we could make this pathway visible by running the reaction in deuterio-acid + $\ce{D2O}$).  Protonation of the benzene ring at the carbon bearing the substituent provides a choice.  Either the proton can again be removed or the stable t-butyl cation can be ejected.  All of these things are happening in the reaction, but only removal of the t-butyl group is visible.