For fixed size sampling designs with high entropy it is well known that the variance of the Horvitz-Thompson estimator can be approximated by the Hájek formula. The interest of this asymptotic variance approximation is that it only involves the first order inclusion probabilities of the statistical units. We extend this variance formula when the variable under study is functional and we prove, under general conditions on the regularity of the individual trajectories and the sampling design, that it asymptotically provides a uniformly consistent estimator of the variance function of the Horvitz-Thompson estimator of the mean function. Rates of convergence to the true variance function are given for rejective sampling. We deduce, under conditions on the entropy of the sampling design, that it is possible to build confidence bands whose coverage is asymptotically the desired one via simulation of Gaussian processes whose variance function is given by the Hájek formula. Finally, the accuracy of the proposed variance estimator is evaluated on samples of electricity consumption data measured every half an hour over a period of one week.