In this letter we study how fast the energy density of a quantum gas can increase in time, when the inter-atomic interaction characterized by the $s$-wave scattering length $a_text{s}$ is increased from zero with arbitrary time dependence. We show that, at short time, the energy density can at most increase as $sqrt{t}$, which can be achieved when the time dependence of $a_text{s}$ is also proportional to $sqrt{t}$, and especially, a universal maximum energy growth rate can be reached when $a_text{s}$ varies as $2sqrt{hbar t/(pi m)}$. If $a_text{s}$ varies faster or slower than $sqrt{t}$, it is respectively proximate to the quench process and the adiabatic process, and both result in a slower energy growth rate. These results are obtained by analyzing the short time dynamics of the short-range behavior of the many-body wave function characterized by the contact, and are also confirmed by numerical solving an example of interacting bosons with time-dependent Bogoliubov theory. These results can also be verified experimentally in ultracold atomic gases.
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