Okay, everyone realizes how far away the core of the Earth is even from the bottom of the ocean, yes? The temperature of earth's core has zero effect on the temperature of the ocean.
Fun fact: If you can imagine Planet Earth to the same scale of an apple, the skin of the apple would be the same size as the top of the sky to the bottom of the ocean.
It must have some effect according to the 2nd law of thermodynamics. Anyway, from the NOAA:
Cold, salty water is dense and sinks to the bottom of the ocean while warm water is less dense and rises to the surface. Cold water has a higher density than warm water. Deep water gets colder at depth because cold, salty ocean water sinks to the bottom of the ocean basins.
To a point yes, water's most important property for life anyways it that it's densest at 4 degrees Celsius (39 F). Water at freezing is actually less dense, and floats to the top, hence why lakes, oceans, rivers, really any body of water freezes from the top down (and why ice floats). This ability to stay above freezing and have the protection of ice at the top is probably the biggest reason why water is such a unique and necessary component of life. It's nearly impossible to freeze deep down because you would have to cool it so much, and the thicker the ice above it becomes, the better protected it is from actually getting colder. Add in pressure and geothermal activity, life is quite able to survive at 39F. Big reason why NASA and others think Europa, one of Jupiter's moon might have life, cause while icy on the outside, once you get below it, it's above freezing.
