Actually, it can, thanks to the exponential set.
cvx_begin
variables y x t
{x,t,y} == exponential;
minimize(y);
This works because the exponential set enforces the constraint te^{x/t}\leq y. However, please note and heed the warnings about the successive approximation method used to solve problems involving log/exp/entropy.
If you look at the help for exponential, you’ll see that it mentions a function exp_p. It would seem we had intended to create it, but never did! You will also see that the help text for exponential uses the name exp_cone in the example instead. That’s a documentation error.