We analyze two multi-chord stellar occultations by Pluto that were observed on 2012 July 18th and 2013 May 4th, and respectively monitored from five and six sites. They provide a total of fifteen light curves, 12 of which were used for a simultaneous fit that uses a unique temperature profile, assuming a clear (no haze) and pure N-2 atmosphere, but allowing for a possible pressure variation between the two dates. We find a solution that satisfactorily fits (i.e., within the noise level) all of the 12 light curves, providing atmospheric constraints between similar to 1190 km (pressure similar to 11 mu bar) and similar to 1450 km (pressure similar to 0.1 mu bar) from Pluto's center. Our main results are: (1) the best-fitting temperature profile shows a stratosphere with a strong positive gradient between 1190 km (at 36 K, 11 mu bar) and r = 1215 km (6.0 mu bar), where a temperature maximum of 110 K is reached; above it is a mesosphere with a negative thermal gradient of -0.2 K km(-1) up to similar to 1390 km (0.25 mu bar), where the mesosphere connects itself to a more isothermal upper branch around 81 K; (2) the pressure shows a small (6%) but significant increase (6 sigma level) between the two dates; (3) without a troposphere, Pluto's radius is found to be R-P = 1190 +/- 5 km. Allowing for a troposphere, R-P is constrained to lie between 1168 and 1195 km; and (4) the currently measured CO abundance is too small to explain the mesospheric negative thermal gradient. Cooling by HCN is possible, but only if this species is largely saturated. Alternative explanations like zonal winds or vertical compositional variations of the atmosphere are unable to explain the observed mesospheric negative thermal gradient.