If you can find a way to shade the container without shading the tree, it will be happier. Watering will help with lowering container temps. Typically, the "cold"water coming out of a Houston tap is not less than 75F. If you water once per day, do so in the evening, or the temps will rebound quickly.
As mentioned, the size of the container matters. You could, however, employ a pot in pot method for smaller containers to add mass without substantially increasing the amount of water needed. Put some mulch in the bottom of a much larger (preferably light colored) container. Put the smaller container in the larger one. You want the top of the smaller one just below the top of the larger one. Fill the larger container with mulch. When you water, it just needs to go in the smaller container. The roots will grow into the mulch. If it bothers you, remove it once a month +/- and trim the roots.