I think the pressure here is "defined" by the expression for the stress-energy tensor you wrote down, or equivalently, by $$ p \equiv \frac{1}{3}\left(\frac{v_{\alpha}v_{\beta}}{c^{2}}T^{\alpha\beta}-T^{\alpha}\!{}_{\alpha}\right). $$ My guess is that you probably would have been happier if the author had called this quantity "the pressure in the comoving frame".

I think the pressure here is "defined" by the expression for the stress-energy tensor you wrote down, or equivalently, by $$ p \equiv \frac{1}{3}\left(\frac{v_{\alpha}v_{\beta}}{c^{2}}T^{\alpha\beta}-T^{\alpha}\!{}_{\alpha}\right). $$ My guess is that you probably would have been happier if the author had called this quantity "the pressure in the (locally) comoving frame".