For $p\ge 1$ let $\varphi_p(x)=x^2/2$ if $|x|\le 1$ and $\varphi_p(x)=1/p|x|^p-1/p+1/2$ if $|x|>1$. For a random variable $ξ$ let $τ_{\varphi_p}(ξ)$ denote $\inf\{a\ge 0:\;\forall_{λ\in\mathbb{R}}\; \ln\mathbb{E}\exp(λξ)\le\varphi_p(aλ)\}$; $τ_{\varphi_p}$ is a norm in a space $Sub_{\varphi_p}=\{ξ:\;τ_{\varphi_p}(ξ)<\infty\}$ of $\varphi_p$-subgaussian random variables. We prove that if for a sequence $(ξ_n)\subset Sub_{\varphi_p}$ ($p>1$) there exist positive constants $c$ and $α$ such that for every natural number $n$ the following inequality $τ_{\varphi_p}(\sum_{i=1}^nξ_i)\le cn^{1-α}$ holds then $n^{-1}\sum_{i=1}^nξ_i$ converges almost surely to zero as $n\to\infty$. This result is a generalization of the SLLN for independent subgaussian random variables (Taylor and Hu \cite{TayHu}) to the case of dependent $\varphi_p$-subgaussian random variables.