Not true. Conservation of energy. When the heater is heating the water, it's using energy. 100% of the time, it's also losing energy to the surroundings. The rate at which it loses energy is proportional to the difference in temperature between it and the ambient temperature surrounding it. Any energy lost, obviously has to be replaced. By allowing it to drop to a somewhat lower temperature, it reduces the rate at which it loses energy, thus reduces the total amount of energy required for the next time the hot water is used.
The OP would probably save more money simply by simply installing a blanket for increased thermal insulation. I doubt the amount saved with a timer would make it worthwhile.
This.
Ohm's Law - it also works for heat transfer.
V = I R
electrical potential difference = current * resistance
current = electrical potential difference / resistance
If there's a voltage differential present, the charge will try to push through any resistance in an effort to reach an equilibrium. The rate at which that charge flows is the current. Assuming the resistance is constant, the current is directly proportional to the voltage.
Temperature:
Difference in temperature = heat transfer * thermal resistance
heat transfer = difference in temperature / thermal resistance
Here too, a difference in temperature represents how much the warm material "wants" to push its energy out into its surroundings, through a thermal resistance, in order to reach equilibrium.
Every degree of thermal differential you can eliminate, the less energy loss will occur. The idea of it taking more energy to "catch up" and get back up to temperature is just psychological trickery. So if you want to reduce the amount of energy lost from your heater to the room in the house, your only options are to fiddle with the two variables that equal the heat transfer: Either reduce the difference in temperature (lower the temperature of the water in the heater, or let the room get hot

), or increase the thermal resistance, by improving the insulation around the tank.
The tank's already got pretty good insulation though, and heating up your house to reduce your water heating bill is only a good solution in the winter. So, your only option left is to reduce the water temperature.
But as we've seen here, the cost savings will be minimal.
I've got gas heat, a gas-fueled water heater, and a gas kitchen stove. In the summer, my gas bill is somewhere around $15/month.
Getting into the apartment's "basement" involves lifting a heavy slab of floor, and then carefully twisting and crawling down some rickety railing-free stairs into an
old, dank, dirty, spider-laden basement, and then doing it again when the vacation's over.
The two occasions I
did turn off the water heater helped me to decide not to do it again - besides the fun-filled visit down to Syringer's Dungeon, I also would end up forgetting to turn the heater back up to the normal temperature, and upon going to get a shower, was greeted with cold water from both sides of the faucet. (That gas burner does do a good job of boosting the temperature fairly quickly though.)