Very little. Three of the four methods in the Knox and Douglass paper from the International Journal of Geosciences you attached yield results that are not statistically different from zero rather than negative as they imply. They also choose to only use six years of ARGO-only data to refute a pattern seen over the last 16 years using a much more extensive array (Lyman et al. 2010) published in Nature. Yes, there is a short-term flattening in the Lyman data as well, which is similar to the flattening from approximately 1994-1998. Decadal variability is a common feature of any climate record. Cherry-picking short pieces of records is poor science when addressing anthropogenic climate change. This variability is discussed in Lyman et al. 2010: "These uncertainties are large enough that interannual variations, such as the 2003–2008 flattening, are statistically meaningless." Knox and Douglass (2010) also mis-cited the results of Lyman et al. (2010). All in all, I doubt Knox and Douglass (2010) would have been published in a reputable journal and should be disregarded.

The heat has been absorbed into the ocean. See publications by Syd Levitus such as the following for more information:

ftp://140.90.235.80/pub/data.nodc/woa/PUBLICATIONS/grlheat08.pdf

Your certainty is not shared with later publications. There has been no significant heat increase in the oceans since 2002-2003 according to the ARGO system . Why? And what does that imply for the energy balance? (see attached publication).

Very little. Three of the four methods in the Knox and Douglass paper from the International Journal of Geosciences you attached yield results that are not statistically different from zero rather than negative as they imply. They also choose to only use six years of ARGO-only data to refute a pattern seen over the last 16 years using a much more extensive array (Lyman et al. 2010) published in Nature. Yes, there is a short-term flattening in the Lyman data as well, which is similar to the flattening from approximately 1994-1998. Decadal variability is a common feature of any climate record. Cherry-picking short pieces of records is poor science when addressing anthropogenic climate change. This variability is discussed in Lyman et al. 2010: "These uncertainties are large enough that interannual variations, such as the 2003–2008 flattening, are statistically meaningless." Knox and Douglass (2010) also mis-cited the results of Lyman et al. (2010). All in all, I doubt Knox and Douglass (2010) would have been published in a reputable journal and should be disregarded.

Also, while not a scientific reference, this recent article in the Guardian discusses some of the recent thoughts on this issue:

http://www.theguardian.com/science/2013/jul/22/climate-change-slowdown-warming-oceans?CMP=twt_gu

A short piece on a recent scientific paper on the subject:

http://www.science20.com/news_articles/case_missing_heat_deep_ocean-115926

showing evidence that the deep ocean has recently been experiencing warming at a more important rate than the upper ocean.

There is more specific heat in the first few meters of the ocean than in all of the

120 plus Kilometers of the atmosphere.

Although there seems to be strong evidence showing that the long trend is that the oceans are indeed heating up. The question was "the missing heat": i.e. can it be shown (through observations not models) that the increase in heat in the oceans matches the "missing heat" in the atmosphere? The heat that should have been there according to the forcings in the models. To my knowledge, this has not been done - yet. (Will the ARGO system be able to present this?) So, if there is a discrepancy between the anticipated heat in the oceans (according to current understanding of the climate sensitivity) will this, again, imply that the climate sensitivity has to be revised? Or, are data too sparse yet for any such conclusions?

What would be interesting is to set up the 1925, 1926 Michelson

experiment in exactly the same location, and remeasure the

Air speed of light again to see how the density of the atmosphere changes

with increasing volume of the atmosphere.

Michelson's ( corrected ) smoky air velocity of light should have been

299, 265,581.9754 meters per second with the light making

exactly 8448 one way trips ( 4224 two way trips ) per second over a distance

of 35, 424.42968459 meters ( one way ) to a ridiculous number of digits.

If the density of the atmosphere ( volume and mass ) of the atmosphere

has increased, then the distance to complete the 8448 one way trips

will decrease as the speed of light slows down while going through

a denser medium than in 1925, 1926.

In short, we have a testable Hypothesis. Does the Volume and the

Pressure and Temperature of the Atmosphere increase with time

due to the extraction of carbon based fuels which are burned and released

into the atmosphere as CO2.

The key to the test is to keep the same location, the same station to

station distance, and the same number of one way and two way trips

that Michelson did in his 1925, 1926 experiment for the light.

The air should be less smoky as the open burning is no longer allowed in

California.

Here's another article on the topic, demonstrating that the warming is being transmitted into deeper ocean waters. From Balmaseda et al, 2013, GRL:

"The elusive nature of the post-2004 upper ocean warming has exposed uncertainties in the ocean's role in the Earth's energy budget and transient climate sensitivity. Here we present the time evolution of the global ocean heat content for 1958 through 2009 from a new observation-based reanalysis of the ocean. Volcanic eruptions and El Niño events are identified as sharp cooling events punctuating a long-term ocean warming trend, while heating continues during the recent upper-ocean-warming hiatus, but the heat is absorbed in the deeper ocean. In the last decade, about 30% of the warming has occurred below 700 m, contributing significantly to an acceleration of the warming trend. The warming below 700 m remains even when the Argo observing system is withdrawn although the trends are reduced. Sensitivity experiments illustrate that surface wind variability is largely responsible for the changing ocean heat vertical distribution."

http://onlinelibrary.wiley.com/doi/10.1002/grl.50382/abstract

Balmaseda et al (2013) show the continuation of heat transferred to deeper layers of the oceans. The results are largely in line with Hansen et. al (2005). (See link). I believe Hansen et al captures the main issues very well and gives a nice platform for a discussion of the missing heat. Most of the projections as well as model results given are consistent with the corresponding parts in the various later articles referred to in the answers given in this thread. One should remember though that the "hiatus" was not acknowledged in 2005. Depending on what number are used, there is an imbalance in the climate system of between 0.85 W/m^2 (Hansen et al 2005) and approx 1.2 W/m2 using updated number. This indicate heating in the "pipeline". But since the response time of the climate system is not known, then I would assume it is impossible to know whether the climate system is heating up as anticipated according to most models or if the excess heat simply radiates out in space. My point is that the heating rate of the oceans (including below 700 m) does not seem to be large enough to cover for the missing heat in the atmosphere and the imbalance in the climate system. Am I right in my reasoning here?

http://www.micro.utexas.edu/courses/THOC/Hansen-et-al-2005-Science.pdf

Being retired I do not have access to institutional libraries and hence can not check the temperature data that Hansen et al. have used in their paper that seems to show a rather continuous increase in ocean heat content. I just wonder how representative this data is in reality, because despite thousands on cruises around the world oceans, I find it highly questionable that the data is sufficiently representative for the entire global ocean.

The ARGO data on the other hand shows no increase in heat content: http://oceans.pmel.noaa.gov/

There is an other minor dilemma. It is said that global warming resulting from increasing greenhouse gases is due to increased infra-red back-radiation, Khiel & Trenberth et al in BAMS Vol. 78, No. 2, February 1997.

Now the interesting part is that infra red (heat) radiation is not capable of penetrating beyond a very thin layer at the surface of the oceans. In addition, this warming of the surface would only result in increased evaporation. So how could any additional heat penetrate deep into the oceans. A further very appropriate question would be, why this additional heat has all of a sudden decided to sink into the oceans during the past decade and a half? Too many unanswered questions???

Well, it looks like I was on the right track on the analysis of the "missing heat". 

http://www.nature.com/nclimate/journal/vaop/ncurrent/full/nclimate2387.html#auth-4

This article agrees with me. No contribution to the energy budget from the oceans since 2005. Is it time to at least consider adjusting the climate sensitivity?

https://niclewis.files.wordpress.com/2016/03/briefing-note-on-climate-sensitivity-etc_nic-lewis_mar2016.pdf

http://www.cgd.ucar.edu/cas/Trenberth/website-archive/trenberth.papers-moved/Balmaseda_Trenberth_Kallen_grl_13.pdf

https://www.geos.ed.ac.uk/homes/ghegerl/KH2008.pdf

From Hamit Cans answer I conclude that the ECS and TCR values used in models, needs revising (downwards). The heat content in the ocean cannot explain the lower rate of warming recently.