Elastic stability criteria are widely employed to prove the being of the lattice. Sin'ko and Smirnov have reported the applicable criteria under isotropic pressure and given the equations between the elastic constants ${\tilde{C}}_{ij}$ and $C_{ij}$ under pressure. On this basis, the closed forms of necessary and sufficient conditions for elastic stability in all crystal classes have been presented, which are popular in normal pressure. However, the forms of elastic stability criteria under pressure are still fragmented in various literature studies. Carbon is an element with a rich variety of allotropes, and because of its excellent mechanical and electronic properties, it gains enduring and intense attention, while its phase diagram is poorly known. In order to systematically study the response of various carbon stabilities to pressure and offer some valuable insights into experimental exploration, we derive the total forms of mechanical stability criteria under isotropic pressure and calculate the mechanical stability of 46 carbon allotropes involving seven crystal systems under pressure.