Imagine a box filled with marbles (glass spheres about 1 cm in diameter). If you push down with your hand on the marbles, the force you provide does not move all marbles in the downward direction. Some will move sideways and some will move upward. This is because the marbles being spheres only contact another marble at a single point. That single point of contact does not provide a rigid connection between marbles. That contact point is free to move anywhere as long as the distance between marble centers remains 1 diameter. The same phenomena is taking place in the fluid, transmitting pressure in all directions.
The hydrostatic pressure will depend on the temperature only to the extent that the column of water above the object has decreased density with increased temperature.