We present an analysis of the performance of some standard and optimized explicitly Runge-Kutta schemes that are equipped with CFL-based and error-based time step adaptivity when they are coupled with the relaxation procedure to achieve fully-discrete entropy stability for complex compressible flow simulations. We investigate the performance of the temporal integration algorithms by simulating the flow past the NASA juncture flow model using the in-house KAUST SSDC hp-adaptive collocated entropy stable discontinuous Galerkin solver. In addition, we present a preliminary analysis of the performance of the SSDC framework on the Amazon web service cloud computing. The results indicate that SSDC scales well on the most recent and exotic computing architectures available on the Amazon cloud platform. Our findings might help select a more robust and efficient temporal integration algorithm and guide the choice of the EC2 AWS instances that give the best price and wall-clock-time performance to simulate industrially relevant turbulent flow problems.