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We propose a new reionization probe that uses cosmic microwave background (CMB) observations; the cross-correlation between fluctuations in the CMB optical depth which probes the integrated electron density, δτ, and the Compton y-map which probes the integrated electron pressure. This cross-correlation is much less contaminated than the y-map power spectrum by late-time cluster contributions. In addition, this cross-correlation can constrain the temperature of ionized bubbles while the optical-depth fluctuations and kinetic Sunyaev Zel'dvich effect can not. We measure this new observable using a Planck y-map as well as a map of optical-depth fluctuations that we reconstruct from Planck CMB temperature data. We use our measurements to derive a first CMB only upper limit on the temperature inside ionized bubbles, Tb≲7.0×105 K (2σ). We also present future forecasts, assuming a fiducial model with characteristic reionization bubble size Rb=5 Mpc and Tb=5×104 K. The signal-to-noise ratio of the fiducial cross-correlation using a signal dominated y-map from the Probe of Inflation and Cosmic Origins (PICO) becomes ≃7 with CMB-S4 δτ and ≃13 with CMB-HD δτ. For the fiducial model, we predict that the CMB-HD--PICO cross-correlation should achieve an accurate measurement of the reionization parameters; Tb≃49800-5100+4500 K and Rb≃5.09-0.79+0.66 Mpc. Since the power spectrum of the electron density fluctuations is constrained by the δτ autospectrum, the temperature constraints should be only weakly model dependent on the details of the electron distributions and should be statistically representative of the temperature in ionized bubbles during reionization. This cross-correlation could, therefore, become an important observable for future CMB experiments. [ABSTRACT FROM AUTHOR]