Accelerating, exact, explicit and simple solutions of relativistic hydrodynamics allow for a simple and natural description of highly relativistic p+p collisions. These solutions yield a finite rapidity distribution, thus they lead to an advanced estimate of the initial energy density of high energy collisions. We show that such an advanced estimate yields an initial energy density in $\sqrt{s}=7$ and 8 TeV p+p collisions at LHC around or above the critical energy density from lattice QCD, and a corresponding initial temperature above the critical temperature from QCD and the Hagedorn temperature. This suggests that the collision energy of the LHC corresponds to a large enough initial energy density to create a non-hadronic perfect fluid even in pp collisions. %We also show, that several times the %critical energy density may have been reached in high multiplicity events, hinting at a non-hadronic medium created in %high multiplicity $\sqrt{s}=7$ and 8 TeV p+p collisions.