Perovskites like LaAlO3 (or SrTiO3) undergo displacive structural phase transitions from a cubic crystal to a trigonal (or tetragonal) structure. For many years, the critical exponents in both these types of transitions have been fitted to those of the isotropic three-components Heisenberg model. However, field theoretical calculations showed that the isotropic fixed point (FP) of the renormalization group is unstable, and renormalization group iterations flow either to a cubic FP or to a fluctuation-driven first-order transition. Here we show that these two scenarios correspond to the cubic-to-trigonal and to the cubic-to-tetragonal transitions, respectively. In both cases, the critical behavior is described by slowly varying effective critical exponents, which exhibit universal features. For the trigonal case, we predict a crossover of the effective exponents from their Ising values to their cubic values (which are close to the isotropic ones). For the tetragonal case, the effective exponents can have the isotropic values over a wide temperature range before exhibiting large changes en route to the first-order transition. New renormalization group calculations near the isotropic FP in three dimensions are presented and used to estimate the effective exponents, and dedicated experiments to test these predictions are proposed. Similar predictions apply to cubic magnetic and ferroelectric systems.