When a heat flux Q is applied downward through a sample of liquid 4He near the lambda transition, the helium self organizes such that the gradient in temperature matches the gravity induced gradient in T λ. All the helium in the sample is then at the same reduced temperature tSOC=TSOC-TλT λ and the helium is said to be in the Self-Organized Critical (SOC) state. We have made preliminary measurements of the 4He SOC state specific heat, C∇T(T(Q)). Despite having a cell height of 2.54 cm, our results show no difference between C∇T and the zero-gravity 4He specific heat results of the Lambda Point Experiment (LPE) [J.A. Lipa et al., Phys. Rev. B, 68, 174518 (2003)] over the range 250 to 450 nK below the transition. There is no gravity rounding because the entire sample is at the same reduced temperature tSOC(Q). Closer to T λ the SOC specific heat falls slightly below LPE, reaching a maximum at approximately 50 nK below Tλ, in agreement with theoretical predictions [R. Haussmann, Phys. Rev. B, 60, 12349 (1999)].