Thought the hot water heater was broke so we replaced it. With a new Hot Water heater installed for the last billing cycle we get only ...?

Warm water out only usually means you hooked the in and the out backwards. Yes, there IS a difference. And if you have an unusually high bill then something somewhere is drawing power. $300 is a big jump. I'd eliminate all sources of energy consumption and check the meter. If after everything is shut off, or unplugged, if the electric meter is moving then somewhere you're getting a drain on your power. OR it could be that the heater is wired wrong, and you're creating a short somewhere within the system.

Good luck.

'')

Water heaters don't cost that much to run monthly. You didn't say if you have electric heating and whether there was cold weather during the month. It's impossible to say what the problem is or whether it was properly installed from this perspective. Try checking both meters for stray usage when everything is turned off. Some other answers gave you this good tip, if that doesn't illuminate the problem you may have to turn to a professional.

Water will not run through the water heater unless a valve is open or there is a leak. If you are on city water, turn off all water faucets and go look at the meter. If there is a leak the water meter will be turning. There is a small red arrow on the meter that is a flow indicator. Any amount of water flowing through the meter will cause it to move. Watch it for a minute to see if it is moving. If it does move with everything turned off you now have to find the leak.

Check the t/p valve on the water heater to make sure that it is closing. There should be no water coming from it.

Did you get it wired up correctly?If water was running through the water heater you could hear it when standing along side of it.Check what your thermostat on the heater is set on,maybe it just needs turned up.

It almost has to be an electrical problem.

If you had it professionally installed, call the installing company and have their best service technician come out and check it. If they goofed, they should fix it for free.

Another free bit of service may be to call your utility company. They often check such problems for free and identify the source of the problem. If repairs are needed you can't perform yourself, then you may have to pay for some additional repair service.

Call a local qualified professional electrician. This sounds like the wiring has a fault.

I could use some expert advice,

That is 100% correct. However, until someone inspects your apartment IN PERSON, they CAN'T give expert advice on this issue.

Been there done that, so I got a tankless water heater now I have no problems at all.